library(dplyr)
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library(coefplot)
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library(gridExtra)
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library(tidyverse)
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library(iterators)
library(caret)
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library(parallel)
library(doParallel)
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df <- readr::read_csv("paint_project_train_data.csv", col_names = TRUE)
## Rows: 835 Columns: 8
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Lightness, Saturation
## dbl (6): R, G, B, Hue, response, outcome
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
df %>% glimpse()
## Rows: 835
## Columns: 8
## $ R <dbl> 172, 26, 172, 28, 170, 175, 90, 194, 171, 122, 0, 88, 144, …
## $ G <dbl> 58, 88, 94, 87, 66, 89, 78, 106, 68, 151, 121, 140, 82, 163…
## $ B <dbl> 62, 151, 58, 152, 58, 65, 136, 53, 107, 59, 88, 58, 132, 50…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "bright", "bright", "bright", "bright", "bright", "bright",…
## $ Hue <dbl> 4, 31, 8, 32, 5, 6, 34, 10, 1, 21, 24, 22, 36, 16, 26, 12, …
## $ response <dbl> 12, 10, 16, 10, 11, 16, 10, 19, 14, 25, 14, 19, 14, 38, 15,…
## $ outcome <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,…
#1. Visualize the distributions of variables in the data set.
#1.1 Counts for categorical variables.
ans: Here we calculate the lightness variables and saturation separately, and we also calculate them into each pair with one lightness and saturation.
counts_lightness <- df %>%
count(Lightness)
counts_saturation <- df %>%
count(Saturation)
counts_df <- df %>%
count(Lightness, Saturation)
print(counts_lightness)
## # A tibble: 7 × 2
## Lightness n
## <chr> <int>
## 1 dark 117
## 2 deep 119
## 3 light 120
## 4 midtone 119
## 5 pale 121
## 6 saturated 119
## 7 soft 120
print(counts_saturation)
## # A tibble: 7 × 2
## Saturation n
## <chr> <int>
## 1 bright 126
## 2 gray 83
## 3 muted 126
## 4 neutral 122
## 5 pure 126
## 6 shaded 126
## 7 subdued 126
print(counts_df)
## # A tibble: 49 × 3
## Lightness Saturation n
## <chr> <chr> <int>
## 1 dark bright 18
## 2 dark gray 10
## 3 dark muted 18
## 4 dark neutral 17
## 5 dark pure 18
## 6 dark shaded 18
## 7 dark subdued 18
## 8 deep bright 18
## 9 deep gray 12
## 10 deep muted 18
## # ℹ 39 more rows
#1.2 Histograms or Density plots for continuous variables. Are the distributions Gaussian like?
Acoording to the graphs below, they are not Gaussian like.
ggplot(df, aes(x = response)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of Response", x = "Response")
## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = R)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of R", x = "R")
## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = G)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of G", x = "G")
## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = B)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of B", x = "B")
## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
ggplot(df, aes(x = Hue)) +
geom_density(fill = "lightblue", color = "darkblue", binwidth = 10) +
labs(title = "Density Plot of Hue", x = "Hue")
## Warning in geom_density(fill = "lightblue", color = "darkblue", binwidth = 10):
## Ignoring unknown parameters: `binwidth`
#2.Condition (group) the continuous variables based on the categorical variables. #2.1 Are there differences in continuous variable distributions and continuous variable summary statistics based on categorical variable values?
library(reshape2)
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## smiths
# 使用 melt 函數將 R, G, B 轉換為一個變數
df_melted <- melt(df, id.vars = c("Lightness"), measure.vars = c("R", "G", "B"))
# 繪製盒狀圖
ggplot(df_melted, aes(x = Lightness, y = value, fill = variable)) +
geom_boxplot() +
labs(title = "Boxplot of R, G, B Grouped by Lightness") +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
# 使用ANOVA進行分組比較
model_anova <- aov(R ~ Lightness, data = df)
anova_result <- summary(model_anova)
print(anova_result)
## Df Sum Sq Mean Sq F value Pr(>F)
## Lightness 6 1460742 243457 157.3 <2e-16 ***
## Residuals 828 1281177 1547
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
In your results, the p-value Pr(>F) is less than 0.05 (common significance level), so we reject the null hypothesis. This implies that there is a statistically significant difference between different levels of Lightness.
#2.2 Are there differences in continuous variable distributions and continuous variable summary statistics based on the binary outcome?
summary(df$response)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 6.0 26.0 51.0 48.6 72.0 87.0
# Summary statistics for 'response' based on 'outcome'
summary(df$response[df$outcome == 1]) # Replace 1 with the actual code for the positive outcome
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 6.00 17.00 46.00 45.22 70.00 85.00
summary(df$response[df$outcome == 0]) # Replace 0 with the actual code for the negative outcome
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 6.00 27.00 52.50 49.61 73.00 87.00
the first set of answers focuses on summarizing the overall distribution of a continuous variable, while the second set of answers explores how the distribution of the continuous variable differs based on a binary outcome variable. The second set of answers involves comparing the distribution of the continuous variable between different groups defined by the binary outcome.
#3.Visualize the relationships between the continuous inputs, are they correlated?
mod01 <- lm( response ~ R*G*B*Hue, data = df )
mod01%>% coefplot::coefplot()+
theme(legend.position = 'none')
As we see below all features are not statistically significant except Hue, because Hue doesn’t contain zero.
# 使用散點圖視覺化連續輸入變量之間的關係
pairs(df[, c("R", "G", "B", "Hue")], pch = 16, col = "blue", main = "Scatterplot Matrix")
# 計算相關性矩陣
cor_matrix <- cor(df[, c("R", "G", "B", "Hue")])
# 視覺化相關性矩陣
heatmap(cor_matrix, annot = TRUE, cmap = "Blues", main = "Correlation Matrix")
## Warning in plot.window(...): "annot" is not a graphical parameter
## Warning in plot.window(...): "cmap" is not a graphical parameter
## Warning in plot.xy(xy, type, ...): "annot" is not a graphical parameter
## Warning in plot.xy(xy, type, ...): "cmap" is not a graphical parameter
## Warning in title(...): "annot" is not a graphical parameter
## Warning in title(...): "cmap" is not a graphical parameter
4.Visualize the relationships between the continuous outputs (response and the LOGIT-transformed response, y) with respect to the continuous INPUTS. 4.1Can you identify any clear trends? Do the trends depend on the categorical INPUTS?
dfii <- df %>%
mutate(y = boot::logit( (response - 0) / (100 - 0) ) ) %>%
select(R, G, B,
Lightness, Saturation, Hue,
y)
dfii %>% glimpse()
## Rows: 835
## Columns: 7
## $ R <dbl> 172, 26, 172, 28, 170, 175, 90, 194, 171, 122, 0, 88, 144, …
## $ G <dbl> 58, 88, 94, 87, 66, 89, 78, 106, 68, 151, 121, 140, 82, 163…
## $ B <dbl> 62, 151, 58, 152, 58, 65, 136, 53, 107, 59, 88, 58, 132, 50…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "bright", "bright", "bright", "bright", "bright", "bright",…
## $ Hue <dbl> 4, 31, 8, 32, 5, 6, 34, 10, 1, 21, 24, 22, 36, 16, 26, 12, …
## $ y <dbl> -1.9924302, -2.1972246, -1.6582281, -2.1972246, -2.0907411,…
mod02 <- lm( y ~ R*G*B*Hue, data = dfii )
mod02 %>% coefplot::coefplot()+
theme(legend.position = 'none')
5.How can you visualize the behavior of the binary outcome with respect to the continuous inputs? How can you visualize the behavior of the binary outcome with respect to the categorical INPUTS?
library(reshape2)
# 使用 melt 函數將 R, G, B 轉換為一個變數
df_melted <- melt(df, id.vars = c("outcome"), measure.vars = c("R", "G", "B"))
# 繪製盒狀圖
ggplot(df_melted, aes(x = outcome, y = value, fill = variable)) +
geom_boxplot() +
labs(title = "Boxplot of R, G, B Grouped by outcome") +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
# Create a boxplot for the binary outcome with respect to Lightness and Saturation
ggplot(df, aes(x = Lightness, y = response, fill = factor(outcome))) +
geom_boxplot() +
labs(title = "Boxplot of Response with Respect to Lightness and Outcome", x = "Lightness", y = "Response", fill = "Outcome") +
facet_wrap(~ Saturation, scales = "free", ncol = 2) # Facet by Saturation
Before using more advanced methods, you need to develop a baseline understanding for the behavior of the LOGIT-transformed response as a function of the inputs using linear modeling techniques.
df_standard <- dfii
# Standardization function
standardize <- function(x) {
return ((x - mean(x)) / sd(x))
}
# Apply the function to the variables
df_standard$R <- standardize(dfii$R)
df_standard$G <- standardize(dfii$G)
df_standard$B <- standardize(dfii$B)
df_standard$Hue <- standardize(dfii$Hue)
df_standard$y <- standardize(dfii$y)
df_standard %>% glimpse()
## Rows: 835
## Columns: 7
## $ R <dbl> -0.19790120, -2.74419521, -0.19790120, -2.70931447, -0.2327…
## $ G <dbl> -2.3619736, -1.7631189, -1.6433480, -1.7830807, -2.2022790,…
## $ B <dbl> -1.7994266, -0.1706872, -1.8726283, -0.1523868, -1.8726283,…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "bright", "bright", "bright", "bright", "bright", "bright",…
## $ Hue <dbl> -1.3548215, 1.3198239, -0.9585777, 1.4188848, -1.2557605, -…
## $ y <dbl> -1.5899718, -1.7628901, -1.3077880, -1.7628901, -1.6729807,…
Use lm() to fit linear models. You must use the following: #A1. Intercept-only model – no INPUTS!
data <- data.frame(logit_response = dfii$y)
intercept_only_model <- lm(logit_response ~ 1, data = data)
summary(intercept_only_model)
##
## Call:
## lm(formula = logit_response ~ 1, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.6422 -0.9366 0.1494 1.0538 2.0103
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.10936 0.04099 -2.668 0.00777 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.184 on 834 degrees of freedom
fit_lm_01 <- lm(y ~ 1, data =df_standard)
fit_lm_01 %>% summary()
##
## Call:
## lm(formula = y ~ 1, data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.2309 -0.7908 0.1261 0.8898 1.6974
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.305e-17 3.461e-02 0 1
##
## Residual standard error: 1 on 834 degrees of freedom
#A2. Categorical variables only – linear additive
fit_lm_02 <- lm(y ~ Lightness + Saturation, data = df_standard)
fit_lm_02 %>% summary()
##
## Call:
## lm(formula = y ~ Lightness + Saturation, data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.90968 -0.22814 -0.01335 0.19000 1.35945
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.30251 0.04224 -30.837 < 2e-16 ***
## Lightnessdeep 0.46151 0.04455 10.360 < 2e-16 ***
## Lightnesslight 2.35306 0.04446 52.931 < 2e-16 ***
## Lightnessmidtone 1.51244 0.04455 33.952 < 2e-16 ***
## Lightnesspale 2.66009 0.04437 59.954 < 2e-16 ***
## Lightnesssaturated 1.01152 0.04455 22.707 < 2e-16 ***
## Lightnesssoft 1.98898 0.04446 44.741 < 2e-16 ***
## Saturationgray -0.47209 0.04837 -9.759 < 2e-16 ***
## Saturationmuted -0.15178 0.04310 -3.521 0.000453 ***
## Saturationneutral -0.27411 0.04346 -6.308 4.62e-10 ***
## Saturationpure 0.34271 0.04310 7.951 6.12e-15 ***
## Saturationshaded -0.25504 0.04310 -5.917 4.81e-09 ***
## Saturationsubdued -0.23723 0.04310 -5.504 4.97e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3421 on 822 degrees of freedom
## Multiple R-squared: 0.8846, Adjusted R-squared: 0.8829
## F-statistic: 525.3 on 12 and 822 DF, p-value: < 2.2e-16
library(coefplot)
coefplot(fit_lm_02)
#A3. Continuous variables only – linear additive
fit_lm_03 <- lm(y ~ R + G + B + Hue, data = df_standard)
coefplot(fit_lm_03)
#A4.All categorical and continuous variables – linear additive
fit_lm_04 <- lm(y ~ ., data = df_standard)
coefplot(fit_lm_04)
#A5. Interaction of the categorical inputs with all continuous inputs main effects
fit_lm_05 <- lm(y ~ (Lightness + Saturation) * (R + G + B + Hue), data = df_standard)
coefplot(fit_lm_05)
#A6. Add categorical inputs to all main effect and all pairwise interactions of continuous inputs
fit_lm_06 <- lm(y ~ Lightness + Saturation + (R + G + B + Hue)^2, data = df_standard)
fit_lm_06 %>% summary()
##
## Call:
## lm(formula = y ~ Lightness + Saturation + (R + G + B + Hue)^2,
## data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.236008 -0.038741 -0.003913 0.035839 0.241332
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.030689 0.018823 -1.630 0.103407
## Lightnessdeep 0.043867 0.010872 4.035 5.98e-05 ***
## Lightnesslight -0.067735 0.025544 -2.652 0.008166 **
## Lightnessmidtone -0.085469 0.019500 -4.383 1.32e-05 ***
## Lightnesspale 0.036528 0.027182 1.344 0.179377
## Lightnesssaturated -0.007675 0.015221 -0.504 0.614230
## Lightnesssoft -0.099784 0.022806 -4.375 1.37e-05 ***
## Saturationgray -0.035763 0.011384 -3.142 0.001741 **
## Saturationmuted -0.017965 0.008552 -2.101 0.035971 *
## Saturationneutral -0.044623 0.009441 -4.727 2.69e-06 ***
## Saturationpure 0.025348 0.008919 2.842 0.004597 **
## Saturationshaded -0.034739 0.009117 -3.810 0.000149 ***
## Saturationsubdued -0.038154 0.008793 -4.339 1.61e-05 ***
## R 0.232483 0.006785 34.264 < 2e-16 ***
## G 0.749420 0.009083 82.512 < 2e-16 ***
## B 0.135239 0.007819 17.296 < 2e-16 ***
## Hue -0.004240 0.005137 -0.825 0.409434
## R:G 0.028167 0.004905 5.742 1.32e-08 ***
## R:B -0.010631 0.007314 -1.453 0.146487
## R:Hue -0.041204 0.007204 -5.720 1.50e-08 ***
## G:B 0.053081 0.006202 8.559 < 2e-16 ***
## G:Hue 0.027295 0.008677 3.146 0.001717 **
## B:Hue 0.007130 0.010809 0.660 0.509713
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06666 on 812 degrees of freedom
## Multiple R-squared: 0.9957, Adjusted R-squared: 0.9956
## F-statistic: 8494 on 22 and 812 DF, p-value: < 2.2e-16
coefplot(fit_lm_06)
#A7. Interaction of the categorical inputs with all main effect and all pairwise interactions of continuous inputs
fit_lm_07 <- lm(y ~ (Lightness + Saturation) * (R + G + B + Hue)^2, data = df_standard)
coefplot(fit_lm_07)
#A8. 3 models with basis functions of your choice Try non-linear basis functions based on your EDA.
fit_lm_08 <- lm(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)), data = df_standard)
fit_lm_08 %>% summary()
##
## Call:
## lm(formula = y ~ (Lightness + Saturation) * ((R + G + B + Hue)^2 +
## I(R^2) + I(G^2) + I(B^2) + I(Hue^2)), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.125500 -0.015352 -0.000605 0.015508 0.133380
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.740e-01 1.229e-01 -4.669 3.68e-06 ***
## Lightnessdeep 4.347e-01 1.180e-01 3.683 0.000250 ***
## Lightnesslight 5.684e-01 1.325e-01 4.289 2.07e-05 ***
## Lightnessmidtone 4.396e-01 1.242e-01 3.539 0.000430 ***
## Lightnesspale 1.024e+00 1.769e-01 5.790 1.10e-08 ***
## Lightnesssaturated 4.619e-01 1.207e-01 3.827 0.000142 ***
## Lightnesssoft 4.665e-01 1.248e-01 3.738 0.000202 ***
## Saturationgray -1.993e-02 3.240e-02 -0.615 0.538765
## Saturationmuted -1.097e-02 3.098e-02 -0.354 0.723496
## Saturationneutral -2.988e-03 3.065e-02 -0.098 0.922355
## Saturationpure -5.557e-02 4.475e-02 -1.242 0.214840
## Saturationshaded -6.139e-04 3.087e-02 -0.020 0.984140
## Saturationsubdued -1.784e-02 3.093e-02 -0.577 0.564358
## R 1.631e-01 4.856e-02 3.359 0.000829 ***
## G 4.170e-01 5.547e-02 7.517 1.90e-13 ***
## B -4.034e-01 1.302e-01 -3.097 0.002038 **
## Hue 1.331e-01 6.460e-02 2.060 0.039763 *
## I(R^2) 4.656e-02 9.126e-03 5.102 4.44e-07 ***
## I(G^2) 2.743e-02 2.873e-02 0.955 0.340111
## I(B^2) -1.243e-01 3.887e-02 -3.197 0.001456 **
## I(Hue^2) 2.758e-03 1.805e-02 0.153 0.878584
## R:G -8.968e-02 3.068e-02 -2.923 0.003584 **
## R:B -2.549e-02 2.682e-02 -0.950 0.342234
## R:Hue 2.807e-02 3.164e-02 0.887 0.375308
## G:B -1.531e-01 4.267e-02 -3.589 0.000358 ***
## G:Hue 3.767e-02 3.407e-02 1.105 0.269365
## B:Hue 8.085e-02 3.987e-02 2.028 0.043005 *
## Lightnessdeep:R -7.683e-03 4.321e-02 -0.178 0.858925
## Lightnesslight:R 1.451e-01 1.371e-01 1.058 0.290250
## Lightnessmidtone:R 5.111e-02 4.755e-02 1.075 0.282878
## Lightnesspale:R -3.972e-01 3.646e-01 -1.089 0.276491
## Lightnesssaturated:R 6.853e-02 4.376e-02 1.566 0.117829
## Lightnesssoft:R 5.585e-02 6.214e-02 0.899 0.369108
## Lightnessdeep:G 2.248e-01 5.025e-02 4.473 9.12e-06 ***
## Lightnesslight:G -2.549e-01 1.660e-01 -1.536 0.125121
## Lightnessmidtone:G 2.251e-01 5.900e-02 3.815 0.000150 ***
## Lightnesspale:G -8.384e-01 4.563e-01 -1.837 0.066623 .
## Lightnesssaturated:G 2.284e-01 5.160e-02 4.427 1.13e-05 ***
## Lightnesssoft:G 7.211e-02 8.030e-02 0.898 0.369491
## Lightnessdeep:B 5.033e-01 1.260e-01 3.996 7.18e-05 ***
## Lightnesslight:B 4.837e-01 1.612e-01 3.001 0.002799 **
## Lightnessmidtone:B 5.347e-01 1.315e-01 4.067 5.36e-05 ***
## Lightnesspale:B 4.947e-01 2.378e-01 2.080 0.037898 *
## Lightnesssaturated:B 5.630e-01 1.260e-01 4.468 9.34e-06 ***
## Lightnesssoft:B 5.238e-01 1.354e-01 3.870 0.000120 ***
## Lightnessdeep:Hue -2.048e-01 6.599e-02 -3.104 0.001994 **
## Lightnesslight:Hue -1.332e-01 7.253e-02 -1.837 0.066663 .
## Lightnessmidtone:Hue -1.589e-01 6.363e-02 -2.497 0.012772 *
## Lightnesspale:Hue -1.625e-01 8.888e-02 -1.828 0.068054 .
## Lightnesssaturated:Hue -1.444e-01 6.403e-02 -2.255 0.024445 *
## Lightnesssoft:Hue -1.594e-01 6.563e-02 -2.429 0.015411 *
## Lightnessdeep:I(R^2) -9.327e-03 8.874e-03 -1.051 0.293647
## Lightnesslight:I(R^2) -7.219e-02 7.377e-02 -0.979 0.328193
## Lightnessmidtone:I(R^2) 4.760e-03 1.779e-02 0.268 0.789132
## Lightnesspale:I(R^2) 2.988e-01 2.231e-01 1.339 0.180897
## Lightnesssaturated:I(R^2) 1.755e-03 9.818e-03 0.179 0.858179
## Lightnesssoft:I(R^2) 1.044e-02 3.702e-02 0.282 0.778111
## Lightnessdeep:I(G^2) 6.690e-02 2.053e-02 3.258 0.001180 **
## Lightnesslight:I(G^2) 3.432e-01 1.293e-01 2.654 0.008158 **
## Lightnessmidtone:I(G^2) 9.108e-02 4.472e-02 2.037 0.042087 *
## Lightnesspale:I(G^2) 2.753e-01 3.361e-01 0.819 0.413091
## Lightnesssaturated:I(G^2) 6.074e-02 2.885e-02 2.105 0.035667 *
## Lightnesssoft:I(G^2) 2.548e-01 7.069e-02 3.605 0.000337 ***
## Lightnessdeep:I(B^2) 1.410e-01 3.725e-02 3.785 0.000168 ***
## Lightnesslight:I(B^2) 2.097e-01 6.587e-02 3.184 0.001523 **
## Lightnessmidtone:I(B^2) 1.675e-01 3.893e-02 4.304 1.94e-05 ***
## Lightnesspale:I(B^2) 2.031e-01 1.088e-01 1.867 0.062397 .
## Lightnesssaturated:I(B^2) 1.667e-01 3.840e-02 4.341 1.65e-05 ***
## Lightnesssoft:I(B^2) 2.116e-01 4.521e-02 4.681 3.49e-06 ***
## Lightnessdeep:I(Hue^2) -3.804e-03 1.196e-02 -0.318 0.750587
## Lightnesslight:I(Hue^2) -4.958e-03 1.680e-02 -0.295 0.768033
## Lightnessmidtone:I(Hue^2) -6.729e-03 1.573e-02 -0.428 0.668964
## Lightnesspale:I(Hue^2) -4.647e-03 1.597e-02 -0.291 0.771202
## Lightnesssaturated:I(Hue^2) -1.369e-02 1.332e-02 -1.028 0.304457
## Lightnesssoft:I(Hue^2) -1.799e-02 1.718e-02 -1.047 0.295356
## Saturationgray:R -7.726e-03 1.336e-01 -0.058 0.953893
## Saturationmuted:R 1.801e-03 2.805e-02 0.064 0.948832
## Saturationneutral:R 6.425e-02 5.264e-02 1.221 0.222646
## Saturationpure:R 3.889e-02 3.024e-02 1.286 0.198924
## Saturationshaded:R -8.517e-03 3.989e-02 -0.214 0.830987
## Saturationsubdued:R 2.550e-02 3.297e-02 0.773 0.439627
## Saturationgray:G -1.178e-01 1.631e-01 -0.723 0.470079
## Saturationmuted:G 6.606e-04 4.410e-02 0.015 0.988053
## Saturationneutral:G -6.066e-02 8.990e-02 -0.675 0.500036
## Saturationpure:G 4.224e-02 3.922e-02 1.077 0.281977
## Saturationshaded:G 3.322e-02 6.555e-02 0.507 0.612488
## Saturationsubdued:G -4.625e-04 5.330e-02 -0.009 0.993079
## Saturationgray:B 1.423e-01 1.037e-01 1.372 0.170585
## Saturationmuted:B 2.640e-02 3.160e-02 0.835 0.403833
## Saturationneutral:B 1.006e-02 5.894e-02 0.171 0.864533
## Saturationpure:B -3.860e-02 2.836e-02 -1.361 0.174060
## Saturationshaded:B -2.382e-02 4.371e-02 -0.545 0.585942
## Saturationsubdued:B 7.099e-04 3.715e-02 0.019 0.984760
## Saturationgray:Hue 3.804e-05 3.719e-02 0.001 0.999184
## Saturationmuted:Hue 2.625e-04 2.964e-02 0.009 0.992935
## Saturationneutral:Hue 4.235e-02 3.060e-02 1.384 0.166833
## Saturationpure:Hue 3.863e-02 3.700e-02 1.044 0.296781
## Saturationshaded:Hue 2.487e-02 3.019e-02 0.824 0.410388
## Saturationsubdued:Hue 1.873e-02 3.018e-02 0.621 0.535115
## Saturationgray:I(R^2) 3.507e-01 4.509e-01 0.778 0.437004
## Saturationmuted:I(R^2) 4.412e-02 1.405e-02 3.140 0.001768 **
## Saturationneutral:I(R^2) -9.845e-02 1.477e-01 -0.667 0.505320
## Saturationpure:I(R^2) 8.689e-03 1.047e-02 0.830 0.406918
## Saturationshaded:I(R^2) -9.684e-02 4.802e-02 -2.016 0.044166 *
## Saturationsubdued:I(R^2) 4.059e-02 2.604e-02 1.559 0.119496
## Saturationgray:I(G^2) 5.604e-02 6.991e-01 0.080 0.936140
## Saturationmuted:I(G^2) 5.587e-02 3.474e-02 1.608 0.108240
## Saturationneutral:I(G^2) -4.113e-01 2.176e-01 -1.890 0.059155 .
## Saturationpure:I(G^2) 1.592e-02 2.779e-02 0.573 0.566951
## Saturationshaded:I(G^2) -6.115e-02 9.074e-02 -0.674 0.500651
## Saturationsubdued:I(G^2) -1.946e-02 4.496e-02 -0.433 0.665365
## Saturationgray:I(B^2) -2.276e-01 2.326e-01 -0.979 0.328175
## Saturationmuted:I(B^2) 5.149e-02 2.271e-02 2.267 0.023712 *
## Saturationneutral:I(B^2) -1.073e-01 7.824e-02 -1.371 0.170714
## Saturationpure:I(B^2) -1.393e-02 1.544e-02 -0.902 0.367315
## Saturationshaded:I(B^2) 4.691e-02 4.523e-02 1.037 0.300067
## Saturationsubdued:I(B^2) 8.180e-03 2.649e-02 0.309 0.757579
## Saturationgray:I(Hue^2) -1.191e-03 1.931e-02 -0.062 0.950860
## Saturationmuted:I(Hue^2) -2.269e-02 1.532e-02 -1.481 0.139129
## Saturationneutral:I(Hue^2) 1.765e-02 1.625e-02 1.086 0.277735
## Saturationpure:I(Hue^2) 1.881e-02 1.399e-02 1.344 0.179297
## Saturationshaded:I(Hue^2) 1.850e-03 1.498e-02 0.124 0.901716
## Saturationsubdued:I(Hue^2) -1.052e-02 1.565e-02 -0.672 0.501866
## Lightnessdeep:R:G 5.004e-03 1.886e-02 0.265 0.790829
## Lightnesslight:R:G 2.533e-01 1.325e-01 1.912 0.056313 .
## Lightnessmidtone:R:G 1.229e-01 4.718e-02 2.605 0.009400 **
## Lightnesspale:R:G 6.957e-01 3.645e-01 1.909 0.056737 .
## Lightnesssaturated:R:G 7.351e-02 2.793e-02 2.632 0.008697 **
## Lightnesssoft:R:G 1.012e-01 7.566e-02 1.337 0.181688
## Lightnessdeep:R:B -1.165e-02 2.386e-02 -0.488 0.625555
## Lightnesslight:R:B -6.598e-02 1.026e-01 -0.643 0.520627
## Lightnessmidtone:R:B 4.224e-03 3.560e-02 0.119 0.905599
## Lightnesspale:R:B -4.629e-01 2.799e-01 -1.654 0.098693 .
## Lightnesssaturated:R:B 4.583e-02 2.789e-02 1.643 0.100854
## Lightnesssoft:R:B 8.404e-03 5.500e-02 0.153 0.878610
## Lightnessdeep:R:Hue -3.360e-03 1.729e-02 -0.194 0.845981
## Lightnesslight:R:Hue 2.510e-03 4.470e-02 0.056 0.955240
## Lightnessmidtone:R:Hue -6.270e-03 2.600e-02 -0.241 0.809552
## Lightnesspale:R:Hue 1.245e-01 7.858e-02 1.585 0.113513
## Lightnesssaturated:R:Hue -1.019e-02 2.032e-02 -0.501 0.616402
## Lightnesssoft:R:Hue -1.562e-03 3.446e-02 -0.045 0.963867
## Lightnessdeep:G:B 1.105e-01 3.088e-02 3.577 0.000373 ***
## Lightnesslight:G:B 2.217e-01 1.222e-01 1.815 0.070063 .
## Lightnessmidtone:G:B 1.522e-01 4.758e-02 3.198 0.001450 **
## Lightnesspale:G:B 5.919e-01 3.194e-01 1.853 0.064293 .
## Lightnesssaturated:G:B 1.181e-01 3.608e-02 3.272 0.001127 **
## Lightnesssoft:G:B 1.260e-01 7.096e-02 1.776 0.076212 .
## Lightnessdeep:G:Hue -8.602e-02 3.038e-02 -2.831 0.004787 **
## Lightnesslight:G:Hue -3.120e-03 5.601e-02 -0.056 0.955598
## Lightnessmidtone:G:Hue -1.028e-02 4.171e-02 -0.247 0.805323
## Lightnesspale:G:Hue -4.457e-02 7.640e-02 -0.583 0.559832
## Lightnesssaturated:G:Hue -2.276e-02 3.461e-02 -0.658 0.511073
## Lightnesssoft:G:Hue -4.206e-02 4.633e-02 -0.908 0.364315
## Lightnessdeep:B:Hue -8.372e-02 2.929e-02 -2.858 0.004397 **
## Lightnesslight:B:Hue -1.382e-01 4.860e-02 -2.844 0.004604 **
## Lightnessmidtone:B:Hue -8.364e-02 3.525e-02 -2.373 0.017953 *
## Lightnesspale:B:Hue -1.631e-01 6.742e-02 -2.418 0.015863 *
## Lightnesssaturated:B:Hue -5.887e-02 3.048e-02 -1.931 0.053880 .
## Lightnesssoft:B:Hue -8.644e-02 4.071e-02 -2.123 0.034103 *
## Saturationgray:R:G -7.067e-01 1.068e+00 -0.662 0.508266
## Saturationmuted:R:G -1.153e-01 3.770e-02 -3.059 0.002316 **
## Saturationneutral:R:G 3.978e-01 3.232e-01 1.231 0.218819
## Saturationpure:R:G 3.260e-06 3.163e-02 0.000 0.999918
## Saturationshaded:R:G 1.796e-01 1.030e-01 1.743 0.081838 .
## Saturationsubdued:R:G -7.833e-02 5.576e-02 -1.405 0.160537
## Saturationgray:R:B 2.204e-01 4.825e-01 0.457 0.647921
## Saturationmuted:R:B 5.159e-02 3.079e-02 1.676 0.094308 .
## Saturationneutral:R:B -2.422e-01 1.320e-01 -1.836 0.066895 .
## Saturationpure:R:B -1.737e-02 2.142e-02 -0.811 0.417748
## Saturationshaded:R:B -4.993e-02 7.533e-02 -0.663 0.507664
## Saturationsubdued:R:B 1.566e-02 3.860e-02 0.406 0.684978
## Saturationgray:R:Hue 5.358e-02 1.348e-01 0.397 0.691182
## Saturationmuted:R:Hue -8.457e-03 3.214e-02 -0.263 0.792529
## Saturationneutral:R:Hue 3.524e-02 7.918e-02 0.445 0.656472
## Saturationpure:R:Hue -2.235e-02 3.801e-02 -0.588 0.556769
## Saturationshaded:R:Hue -4.133e-02 4.705e-02 -0.878 0.380105
## Saturationsubdued:R:Hue -2.503e-02 3.787e-02 -0.661 0.508859
## Saturationgray:G:B 3.230e-01 6.893e-01 0.469 0.639517
## Saturationmuted:G:B -7.810e-02 5.265e-02 -1.483 0.138482
## Saturationneutral:G:B 4.554e-01 2.291e-01 1.987 0.047300 *
## Saturationpure:G:B 1.353e-02 3.454e-02 0.392 0.695318
## Saturationshaded:G:B -3.276e-02 1.269e-01 -0.258 0.796315
## Saturationsubdued:G:B 4.067e-02 6.306e-02 0.645 0.519252
## Saturationgray:G:Hue -1.215e-01 1.564e-01 -0.777 0.437700
## Saturationmuted:G:Hue -1.850e-02 2.909e-02 -0.636 0.525012
## Saturationneutral:G:Hue -1.792e-02 7.618e-02 -0.235 0.814089
## Saturationpure:G:Hue 9.135e-03 2.725e-02 0.335 0.737552
## Saturationshaded:G:Hue 7.314e-02 4.226e-02 1.731 0.083967 .
## Saturationsubdued:G:Hue -3.253e-02 3.569e-02 -0.911 0.362429
## Saturationgray:B:Hue 7.069e-02 9.546e-02 0.741 0.459213
## Saturationmuted:B:Hue 1.980e-02 3.904e-02 0.507 0.612201
## Saturationneutral:B:Hue -4.266e-02 5.169e-02 -0.825 0.409520
## Saturationpure:B:Hue -7.908e-03 3.794e-02 -0.208 0.834956
## Saturationshaded:B:Hue -5.420e-02 4.629e-02 -1.171 0.242119
## Saturationsubdued:B:Hue 2.367e-02 4.151e-02 0.570 0.568665
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03175 on 640 degrees of freedom
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.999
## F-statistic: 4260 on 194 and 640 DF, p-value: < 2.2e-16
#A9. Can consider interactions of basis functions with other basis functions!
fit_lm_09 <- lm(y ~ Lightness + Saturation + R*G*B*Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2), data = df_standard)
fit_lm_09 %>% summary()
##
## Call:
## lm(formula = y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) +
## I(G^2) + I(B^2) + I(Hue^2), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.197582 -0.027025 -0.002384 0.026201 0.250074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.1657261 0.0174878 -9.477 < 2e-16 ***
## Lightnessdeep 0.0353333 0.0083990 4.207 2.88e-05 ***
## Lightnesslight 0.0058840 0.0192118 0.306 0.759480
## Lightnessmidtone 0.0456371 0.0161270 2.830 0.004773 **
## Lightnesspale 0.0573979 0.0199970 2.870 0.004208 **
## Lightnesssaturated 0.0502504 0.0125656 3.999 6.95e-05 ***
## Lightnesssoft 0.0190397 0.0179929 1.058 0.290291
## Saturationgray 0.0054304 0.0082281 0.660 0.509455
## Saturationmuted 0.0081681 0.0059433 1.374 0.169723
## Saturationneutral -0.0018902 0.0068573 -0.276 0.782886
## Saturationpure 0.0217553 0.0063061 3.450 0.000590 ***
## Saturationshaded 0.0036000 0.0065442 0.550 0.582405
## Saturationsubdued 0.0007582 0.0062741 0.121 0.903848
## R 0.2396113 0.0073988 32.385 < 2e-16 ***
## G 0.6455037 0.0106439 60.645 < 2e-16 ***
## B 0.1285391 0.0097490 13.185 < 2e-16 ***
## Hue 0.0129430 0.0057774 2.240 0.025346 *
## I(R^2) 0.0490690 0.0042808 11.463 < 2e-16 ***
## I(G^2) 0.0957508 0.0078753 12.158 < 2e-16 ***
## I(B^2) 0.0396873 0.0059323 6.690 4.18e-11 ***
## I(Hue^2) -0.0162166 0.0033058 -4.906 1.13e-06 ***
## R:G 0.0177352 0.0085847 2.066 0.039157 *
## R:B -0.0051333 0.0073410 -0.699 0.484585
## G:B -0.0123909 0.0100415 -1.234 0.217575
## R:Hue 0.0076914 0.0086092 0.893 0.371914
## G:Hue 0.0245189 0.0071463 3.431 0.000632 ***
## B:Hue -0.0318680 0.0096608 -3.299 0.001014 **
## R:G:B 0.0994143 0.0043243 22.990 < 2e-16 ***
## R:G:Hue -0.0476393 0.0067120 -7.098 2.79e-12 ***
## R:B:Hue 0.0116556 0.0082037 1.421 0.155770
## G:B:Hue 0.0313149 0.0051365 6.097 1.68e-09 ***
## R:G:B:Hue -0.0055666 0.0032905 -1.692 0.091092 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04481 on 803 degrees of freedom
## Multiple R-squared: 0.9981, Adjusted R-squared: 0.998
## F-statistic: 1.338e+04 on 31 and 803 DF, p-value: < 2.2e-16
#A10. Can consider interactions of basis functions with the categorical inputs!
fit_lm_10 <- lm(y ~ (Lightness + Saturation) * R*G*B*Hue, data = df_standard)
##save model
fit_lm_01 %>% readr::write_rds("mod01.rds")
fit_lm_02 %>% readr::write_rds("mod02.rds")
fit_lm_03 %>% readr::write_rds("mod03.rds")
fit_lm_04 %>% readr::write_rds("mod04.rds")
fit_lm_05 %>% readr::write_rds("mod05.rds")
fit_lm_06 %>% readr::write_rds("mod06.rds")
fit_lm_07 %>% readr::write_rds("mod07.rds")
fit_lm_08 %>% readr::write_rds("mod08.rds")
fit_lm_09 %>% readr::write_rds("mod09.rds")
fit_lm_10 %>% readr::write_rds("mod10.rds")
##reload model
re_load_mod01 <- readr::read_rds("mod01.rds")
re_load_mod02 <- readr::read_rds("mod02.rds")
re_load_mod03 <- readr::read_rds("mod03.rds")
re_load_mod04 <- readr::read_rds("mod04.rds")
re_load_mod05 <- readr::read_rds("mod05.rds")
re_load_mod06 <- readr::read_rds("mod06.rds")
re_load_mod07 <- readr::read_rds("mod07.rds")
re_load_mod08 <- readr::read_rds("mod08.rds")
re_load_mod09 <- readr::read_rds("mod09.rds")
re_load_mod10 <- readr::read_rds("mod10.rds")
#1. Which of the 10 models is the best? What performance metric did you use to make your selection?
extract_metrics <- function(mod, mod_name)
{
broom::glance(mod) %>% mutate(mod_name = mod_name)
}
all_metrics <- purrr::map2_dfr(list(re_load_mod01, re_load_mod02, re_load_mod03, re_load_mod04, re_load_mod05, re_load_mod06, re_load_mod07, re_load_mod08, re_load_mod09, re_load_mod10),
sprintf("%02d", 1:10),
extract_metrics)
all_metrics %>% glimpse()
## Rows: 10
## Columns: 13
## $ r.squared <dbl> 0.0000000, 0.8846326, 0.9881038, 0.9945082, 0.9978083, 0…
## $ adj.r.squared <dbl> 0.0000000, 0.8829484, 0.9880465, 0.9944008, 0.9976262, 0…
## $ sigma <dbl> 1.00000000, 0.34212801, 0.10933222, 0.07482803, 0.048722…
## $ statistic <dbl> NA, 525.2554, 17235.0403, 9258.1844, 5477.4752, 8494.316…
## $ p.value <dbl> NA, 0, 0, 0, 0, 0, 0, 0, 0, 0
## $ df <dbl> NA, 12, 4, 16, 64, 22, 142, 194, 31, 207
## $ logLik <dbl> -1184.3134, -282.6663, 665.8529, 988.5639, 1372.0751, 10…
## $ AIC <dbl> 2372.6268, 593.3327, -1319.7058, -1941.1279, -2612.1502,…
## $ BIC <dbl> 2382.0816, 659.5167, -1291.3412, -1856.0341, -2300.1397,…
## $ deviance <dbl> 834.0000000, 96.2163935, 9.9214333, 4.5801732, 1.8278624…
## $ df.residual <int> 834, 822, 830, 818, 770, 812, 692, 640, 803, 627
## $ nobs <int> 835, 835, 835, 835, 835, 835, 835, 835, 835, 835
## $ mod_name <chr> "01", "02", "03", "04", "05", "06", "07", "08", "09", "1…
#2. Visualize the coefficient summaries for your top 3 models.
all_metrics %>%
select(mod_name, AIC, BIC) %>%
pivot_longer(c(AIC, BIC)) %>%
ggplot(mapping = aes(x = mod_name, y = value)) +
geom_point(size = 2) +
facet_wrap(~name, scales = 'free_y') +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Based on the information provided, model 8 is selected as the optimal model by the less conservative Akaike Information Criterion (AIC), while the more conservative Bayesian Information Criterion (BIC) indicates that model 9 is preferable. Given that my evaluation criteria prioritize BIC as the performance metric, I conclude that model 9 is the superior model.
#3. How do the coefficient summaries compare between the top 3 models?
Model 9 considers the smallest set of features, whereas model 8 incorporates the broadest range. Both models concur on the significance of certain features associated with G.
• Which inputs seem important? In examining the summaries of models 9, 8, and 5, it appears that the features related to the continuous variable G are of significance.
re_load_mod09 %>% coefplot::coefplot()
re_load_mod08 %>% coefplot::coefplot()
re_load_mod05 %>% coefplot::coefplot()
re_load_mod09 %>% summary()
##
## Call:
## lm(formula = y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) +
## I(G^2) + I(B^2) + I(Hue^2), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.197582 -0.027025 -0.002384 0.026201 0.250074
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.1657261 0.0174878 -9.477 < 2e-16 ***
## Lightnessdeep 0.0353333 0.0083990 4.207 2.88e-05 ***
## Lightnesslight 0.0058840 0.0192118 0.306 0.759480
## Lightnessmidtone 0.0456371 0.0161270 2.830 0.004773 **
## Lightnesspale 0.0573979 0.0199970 2.870 0.004208 **
## Lightnesssaturated 0.0502504 0.0125656 3.999 6.95e-05 ***
## Lightnesssoft 0.0190397 0.0179929 1.058 0.290291
## Saturationgray 0.0054304 0.0082281 0.660 0.509455
## Saturationmuted 0.0081681 0.0059433 1.374 0.169723
## Saturationneutral -0.0018902 0.0068573 -0.276 0.782886
## Saturationpure 0.0217553 0.0063061 3.450 0.000590 ***
## Saturationshaded 0.0036000 0.0065442 0.550 0.582405
## Saturationsubdued 0.0007582 0.0062741 0.121 0.903848
## R 0.2396113 0.0073988 32.385 < 2e-16 ***
## G 0.6455037 0.0106439 60.645 < 2e-16 ***
## B 0.1285391 0.0097490 13.185 < 2e-16 ***
## Hue 0.0129430 0.0057774 2.240 0.025346 *
## I(R^2) 0.0490690 0.0042808 11.463 < 2e-16 ***
## I(G^2) 0.0957508 0.0078753 12.158 < 2e-16 ***
## I(B^2) 0.0396873 0.0059323 6.690 4.18e-11 ***
## I(Hue^2) -0.0162166 0.0033058 -4.906 1.13e-06 ***
## R:G 0.0177352 0.0085847 2.066 0.039157 *
## R:B -0.0051333 0.0073410 -0.699 0.484585
## G:B -0.0123909 0.0100415 -1.234 0.217575
## R:Hue 0.0076914 0.0086092 0.893 0.371914
## G:Hue 0.0245189 0.0071463 3.431 0.000632 ***
## B:Hue -0.0318680 0.0096608 -3.299 0.001014 **
## R:G:B 0.0994143 0.0043243 22.990 < 2e-16 ***
## R:G:Hue -0.0476393 0.0067120 -7.098 2.79e-12 ***
## R:B:Hue 0.0116556 0.0082037 1.421 0.155770
## G:B:Hue 0.0313149 0.0051365 6.097 1.68e-09 ***
## R:G:B:Hue -0.0055666 0.0032905 -1.692 0.091092 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04481 on 803 degrees of freedom
## Multiple R-squared: 0.9981, Adjusted R-squared: 0.998
## F-statistic: 1.338e+04 on 31 and 803 DF, p-value: < 2.2e-16
re_load_mod08 %>% summary()
##
## Call:
## lm(formula = y ~ (Lightness + Saturation) * ((R + G + B + Hue)^2 +
## I(R^2) + I(G^2) + I(B^2) + I(Hue^2)), data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.125500 -0.015352 -0.000605 0.015508 0.133380
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.740e-01 1.229e-01 -4.669 3.68e-06 ***
## Lightnessdeep 4.347e-01 1.180e-01 3.683 0.000250 ***
## Lightnesslight 5.684e-01 1.325e-01 4.289 2.07e-05 ***
## Lightnessmidtone 4.396e-01 1.242e-01 3.539 0.000430 ***
## Lightnesspale 1.024e+00 1.769e-01 5.790 1.10e-08 ***
## Lightnesssaturated 4.619e-01 1.207e-01 3.827 0.000142 ***
## Lightnesssoft 4.665e-01 1.248e-01 3.738 0.000202 ***
## Saturationgray -1.993e-02 3.240e-02 -0.615 0.538765
## Saturationmuted -1.097e-02 3.098e-02 -0.354 0.723496
## Saturationneutral -2.988e-03 3.065e-02 -0.098 0.922355
## Saturationpure -5.557e-02 4.475e-02 -1.242 0.214840
## Saturationshaded -6.139e-04 3.087e-02 -0.020 0.984140
## Saturationsubdued -1.784e-02 3.093e-02 -0.577 0.564358
## R 1.631e-01 4.856e-02 3.359 0.000829 ***
## G 4.170e-01 5.547e-02 7.517 1.90e-13 ***
## B -4.034e-01 1.302e-01 -3.097 0.002038 **
## Hue 1.331e-01 6.460e-02 2.060 0.039763 *
## I(R^2) 4.656e-02 9.126e-03 5.102 4.44e-07 ***
## I(G^2) 2.743e-02 2.873e-02 0.955 0.340111
## I(B^2) -1.243e-01 3.887e-02 -3.197 0.001456 **
## I(Hue^2) 2.758e-03 1.805e-02 0.153 0.878584
## R:G -8.968e-02 3.068e-02 -2.923 0.003584 **
## R:B -2.549e-02 2.682e-02 -0.950 0.342234
## R:Hue 2.807e-02 3.164e-02 0.887 0.375308
## G:B -1.531e-01 4.267e-02 -3.589 0.000358 ***
## G:Hue 3.767e-02 3.407e-02 1.105 0.269365
## B:Hue 8.085e-02 3.987e-02 2.028 0.043005 *
## Lightnessdeep:R -7.683e-03 4.321e-02 -0.178 0.858925
## Lightnesslight:R 1.451e-01 1.371e-01 1.058 0.290250
## Lightnessmidtone:R 5.111e-02 4.755e-02 1.075 0.282878
## Lightnesspale:R -3.972e-01 3.646e-01 -1.089 0.276491
## Lightnesssaturated:R 6.853e-02 4.376e-02 1.566 0.117829
## Lightnesssoft:R 5.585e-02 6.214e-02 0.899 0.369108
## Lightnessdeep:G 2.248e-01 5.025e-02 4.473 9.12e-06 ***
## Lightnesslight:G -2.549e-01 1.660e-01 -1.536 0.125121
## Lightnessmidtone:G 2.251e-01 5.900e-02 3.815 0.000150 ***
## Lightnesspale:G -8.384e-01 4.563e-01 -1.837 0.066623 .
## Lightnesssaturated:G 2.284e-01 5.160e-02 4.427 1.13e-05 ***
## Lightnesssoft:G 7.211e-02 8.030e-02 0.898 0.369491
## Lightnessdeep:B 5.033e-01 1.260e-01 3.996 7.18e-05 ***
## Lightnesslight:B 4.837e-01 1.612e-01 3.001 0.002799 **
## Lightnessmidtone:B 5.347e-01 1.315e-01 4.067 5.36e-05 ***
## Lightnesspale:B 4.947e-01 2.378e-01 2.080 0.037898 *
## Lightnesssaturated:B 5.630e-01 1.260e-01 4.468 9.34e-06 ***
## Lightnesssoft:B 5.238e-01 1.354e-01 3.870 0.000120 ***
## Lightnessdeep:Hue -2.048e-01 6.599e-02 -3.104 0.001994 **
## Lightnesslight:Hue -1.332e-01 7.253e-02 -1.837 0.066663 .
## Lightnessmidtone:Hue -1.589e-01 6.363e-02 -2.497 0.012772 *
## Lightnesspale:Hue -1.625e-01 8.888e-02 -1.828 0.068054 .
## Lightnesssaturated:Hue -1.444e-01 6.403e-02 -2.255 0.024445 *
## Lightnesssoft:Hue -1.594e-01 6.563e-02 -2.429 0.015411 *
## Lightnessdeep:I(R^2) -9.327e-03 8.874e-03 -1.051 0.293647
## Lightnesslight:I(R^2) -7.219e-02 7.377e-02 -0.979 0.328193
## Lightnessmidtone:I(R^2) 4.760e-03 1.779e-02 0.268 0.789132
## Lightnesspale:I(R^2) 2.988e-01 2.231e-01 1.339 0.180897
## Lightnesssaturated:I(R^2) 1.755e-03 9.818e-03 0.179 0.858179
## Lightnesssoft:I(R^2) 1.044e-02 3.702e-02 0.282 0.778111
## Lightnessdeep:I(G^2) 6.690e-02 2.053e-02 3.258 0.001180 **
## Lightnesslight:I(G^2) 3.432e-01 1.293e-01 2.654 0.008158 **
## Lightnessmidtone:I(G^2) 9.108e-02 4.472e-02 2.037 0.042087 *
## Lightnesspale:I(G^2) 2.753e-01 3.361e-01 0.819 0.413091
## Lightnesssaturated:I(G^2) 6.074e-02 2.885e-02 2.105 0.035667 *
## Lightnesssoft:I(G^2) 2.548e-01 7.069e-02 3.605 0.000337 ***
## Lightnessdeep:I(B^2) 1.410e-01 3.725e-02 3.785 0.000168 ***
## Lightnesslight:I(B^2) 2.097e-01 6.587e-02 3.184 0.001523 **
## Lightnessmidtone:I(B^2) 1.675e-01 3.893e-02 4.304 1.94e-05 ***
## Lightnesspale:I(B^2) 2.031e-01 1.088e-01 1.867 0.062397 .
## Lightnesssaturated:I(B^2) 1.667e-01 3.840e-02 4.341 1.65e-05 ***
## Lightnesssoft:I(B^2) 2.116e-01 4.521e-02 4.681 3.49e-06 ***
## Lightnessdeep:I(Hue^2) -3.804e-03 1.196e-02 -0.318 0.750587
## Lightnesslight:I(Hue^2) -4.958e-03 1.680e-02 -0.295 0.768033
## Lightnessmidtone:I(Hue^2) -6.729e-03 1.573e-02 -0.428 0.668964
## Lightnesspale:I(Hue^2) -4.647e-03 1.597e-02 -0.291 0.771202
## Lightnesssaturated:I(Hue^2) -1.369e-02 1.332e-02 -1.028 0.304457
## Lightnesssoft:I(Hue^2) -1.799e-02 1.718e-02 -1.047 0.295356
## Saturationgray:R -7.726e-03 1.336e-01 -0.058 0.953893
## Saturationmuted:R 1.801e-03 2.805e-02 0.064 0.948832
## Saturationneutral:R 6.425e-02 5.264e-02 1.221 0.222646
## Saturationpure:R 3.889e-02 3.024e-02 1.286 0.198924
## Saturationshaded:R -8.517e-03 3.989e-02 -0.214 0.830987
## Saturationsubdued:R 2.550e-02 3.297e-02 0.773 0.439627
## Saturationgray:G -1.178e-01 1.631e-01 -0.723 0.470079
## Saturationmuted:G 6.606e-04 4.410e-02 0.015 0.988053
## Saturationneutral:G -6.066e-02 8.990e-02 -0.675 0.500036
## Saturationpure:G 4.224e-02 3.922e-02 1.077 0.281977
## Saturationshaded:G 3.322e-02 6.555e-02 0.507 0.612488
## Saturationsubdued:G -4.625e-04 5.330e-02 -0.009 0.993079
## Saturationgray:B 1.423e-01 1.037e-01 1.372 0.170585
## Saturationmuted:B 2.640e-02 3.160e-02 0.835 0.403833
## Saturationneutral:B 1.006e-02 5.894e-02 0.171 0.864533
## Saturationpure:B -3.860e-02 2.836e-02 -1.361 0.174060
## Saturationshaded:B -2.382e-02 4.371e-02 -0.545 0.585942
## Saturationsubdued:B 7.099e-04 3.715e-02 0.019 0.984760
## Saturationgray:Hue 3.804e-05 3.719e-02 0.001 0.999184
## Saturationmuted:Hue 2.625e-04 2.964e-02 0.009 0.992935
## Saturationneutral:Hue 4.235e-02 3.060e-02 1.384 0.166833
## Saturationpure:Hue 3.863e-02 3.700e-02 1.044 0.296781
## Saturationshaded:Hue 2.487e-02 3.019e-02 0.824 0.410388
## Saturationsubdued:Hue 1.873e-02 3.018e-02 0.621 0.535115
## Saturationgray:I(R^2) 3.507e-01 4.509e-01 0.778 0.437004
## Saturationmuted:I(R^2) 4.412e-02 1.405e-02 3.140 0.001768 **
## Saturationneutral:I(R^2) -9.845e-02 1.477e-01 -0.667 0.505320
## Saturationpure:I(R^2) 8.689e-03 1.047e-02 0.830 0.406918
## Saturationshaded:I(R^2) -9.684e-02 4.802e-02 -2.016 0.044166 *
## Saturationsubdued:I(R^2) 4.059e-02 2.604e-02 1.559 0.119496
## Saturationgray:I(G^2) 5.604e-02 6.991e-01 0.080 0.936140
## Saturationmuted:I(G^2) 5.587e-02 3.474e-02 1.608 0.108240
## Saturationneutral:I(G^2) -4.113e-01 2.176e-01 -1.890 0.059155 .
## Saturationpure:I(G^2) 1.592e-02 2.779e-02 0.573 0.566951
## Saturationshaded:I(G^2) -6.115e-02 9.074e-02 -0.674 0.500651
## Saturationsubdued:I(G^2) -1.946e-02 4.496e-02 -0.433 0.665365
## Saturationgray:I(B^2) -2.276e-01 2.326e-01 -0.979 0.328175
## Saturationmuted:I(B^2) 5.149e-02 2.271e-02 2.267 0.023712 *
## Saturationneutral:I(B^2) -1.073e-01 7.824e-02 -1.371 0.170714
## Saturationpure:I(B^2) -1.393e-02 1.544e-02 -0.902 0.367315
## Saturationshaded:I(B^2) 4.691e-02 4.523e-02 1.037 0.300067
## Saturationsubdued:I(B^2) 8.180e-03 2.649e-02 0.309 0.757579
## Saturationgray:I(Hue^2) -1.191e-03 1.931e-02 -0.062 0.950860
## Saturationmuted:I(Hue^2) -2.269e-02 1.532e-02 -1.481 0.139129
## Saturationneutral:I(Hue^2) 1.765e-02 1.625e-02 1.086 0.277735
## Saturationpure:I(Hue^2) 1.881e-02 1.399e-02 1.344 0.179297
## Saturationshaded:I(Hue^2) 1.850e-03 1.498e-02 0.124 0.901716
## Saturationsubdued:I(Hue^2) -1.052e-02 1.565e-02 -0.672 0.501866
## Lightnessdeep:R:G 5.004e-03 1.886e-02 0.265 0.790829
## Lightnesslight:R:G 2.533e-01 1.325e-01 1.912 0.056313 .
## Lightnessmidtone:R:G 1.229e-01 4.718e-02 2.605 0.009400 **
## Lightnesspale:R:G 6.957e-01 3.645e-01 1.909 0.056737 .
## Lightnesssaturated:R:G 7.351e-02 2.793e-02 2.632 0.008697 **
## Lightnesssoft:R:G 1.012e-01 7.566e-02 1.337 0.181688
## Lightnessdeep:R:B -1.165e-02 2.386e-02 -0.488 0.625555
## Lightnesslight:R:B -6.598e-02 1.026e-01 -0.643 0.520627
## Lightnessmidtone:R:B 4.224e-03 3.560e-02 0.119 0.905599
## Lightnesspale:R:B -4.629e-01 2.799e-01 -1.654 0.098693 .
## Lightnesssaturated:R:B 4.583e-02 2.789e-02 1.643 0.100854
## Lightnesssoft:R:B 8.404e-03 5.500e-02 0.153 0.878610
## Lightnessdeep:R:Hue -3.360e-03 1.729e-02 -0.194 0.845981
## Lightnesslight:R:Hue 2.510e-03 4.470e-02 0.056 0.955240
## Lightnessmidtone:R:Hue -6.270e-03 2.600e-02 -0.241 0.809552
## Lightnesspale:R:Hue 1.245e-01 7.858e-02 1.585 0.113513
## Lightnesssaturated:R:Hue -1.019e-02 2.032e-02 -0.501 0.616402
## Lightnesssoft:R:Hue -1.562e-03 3.446e-02 -0.045 0.963867
## Lightnessdeep:G:B 1.105e-01 3.088e-02 3.577 0.000373 ***
## Lightnesslight:G:B 2.217e-01 1.222e-01 1.815 0.070063 .
## Lightnessmidtone:G:B 1.522e-01 4.758e-02 3.198 0.001450 **
## Lightnesspale:G:B 5.919e-01 3.194e-01 1.853 0.064293 .
## Lightnesssaturated:G:B 1.181e-01 3.608e-02 3.272 0.001127 **
## Lightnesssoft:G:B 1.260e-01 7.096e-02 1.776 0.076212 .
## Lightnessdeep:G:Hue -8.602e-02 3.038e-02 -2.831 0.004787 **
## Lightnesslight:G:Hue -3.120e-03 5.601e-02 -0.056 0.955598
## Lightnessmidtone:G:Hue -1.028e-02 4.171e-02 -0.247 0.805323
## Lightnesspale:G:Hue -4.457e-02 7.640e-02 -0.583 0.559832
## Lightnesssaturated:G:Hue -2.276e-02 3.461e-02 -0.658 0.511073
## Lightnesssoft:G:Hue -4.206e-02 4.633e-02 -0.908 0.364315
## Lightnessdeep:B:Hue -8.372e-02 2.929e-02 -2.858 0.004397 **
## Lightnesslight:B:Hue -1.382e-01 4.860e-02 -2.844 0.004604 **
## Lightnessmidtone:B:Hue -8.364e-02 3.525e-02 -2.373 0.017953 *
## Lightnesspale:B:Hue -1.631e-01 6.742e-02 -2.418 0.015863 *
## Lightnesssaturated:B:Hue -5.887e-02 3.048e-02 -1.931 0.053880 .
## Lightnesssoft:B:Hue -8.644e-02 4.071e-02 -2.123 0.034103 *
## Saturationgray:R:G -7.067e-01 1.068e+00 -0.662 0.508266
## Saturationmuted:R:G -1.153e-01 3.770e-02 -3.059 0.002316 **
## Saturationneutral:R:G 3.978e-01 3.232e-01 1.231 0.218819
## Saturationpure:R:G 3.260e-06 3.163e-02 0.000 0.999918
## Saturationshaded:R:G 1.796e-01 1.030e-01 1.743 0.081838 .
## Saturationsubdued:R:G -7.833e-02 5.576e-02 -1.405 0.160537
## Saturationgray:R:B 2.204e-01 4.825e-01 0.457 0.647921
## Saturationmuted:R:B 5.159e-02 3.079e-02 1.676 0.094308 .
## Saturationneutral:R:B -2.422e-01 1.320e-01 -1.836 0.066895 .
## Saturationpure:R:B -1.737e-02 2.142e-02 -0.811 0.417748
## Saturationshaded:R:B -4.993e-02 7.533e-02 -0.663 0.507664
## Saturationsubdued:R:B 1.566e-02 3.860e-02 0.406 0.684978
## Saturationgray:R:Hue 5.358e-02 1.348e-01 0.397 0.691182
## Saturationmuted:R:Hue -8.457e-03 3.214e-02 -0.263 0.792529
## Saturationneutral:R:Hue 3.524e-02 7.918e-02 0.445 0.656472
## Saturationpure:R:Hue -2.235e-02 3.801e-02 -0.588 0.556769
## Saturationshaded:R:Hue -4.133e-02 4.705e-02 -0.878 0.380105
## Saturationsubdued:R:Hue -2.503e-02 3.787e-02 -0.661 0.508859
## Saturationgray:G:B 3.230e-01 6.893e-01 0.469 0.639517
## Saturationmuted:G:B -7.810e-02 5.265e-02 -1.483 0.138482
## Saturationneutral:G:B 4.554e-01 2.291e-01 1.987 0.047300 *
## Saturationpure:G:B 1.353e-02 3.454e-02 0.392 0.695318
## Saturationshaded:G:B -3.276e-02 1.269e-01 -0.258 0.796315
## Saturationsubdued:G:B 4.067e-02 6.306e-02 0.645 0.519252
## Saturationgray:G:Hue -1.215e-01 1.564e-01 -0.777 0.437700
## Saturationmuted:G:Hue -1.850e-02 2.909e-02 -0.636 0.525012
## Saturationneutral:G:Hue -1.792e-02 7.618e-02 -0.235 0.814089
## Saturationpure:G:Hue 9.135e-03 2.725e-02 0.335 0.737552
## Saturationshaded:G:Hue 7.314e-02 4.226e-02 1.731 0.083967 .
## Saturationsubdued:G:Hue -3.253e-02 3.569e-02 -0.911 0.362429
## Saturationgray:B:Hue 7.069e-02 9.546e-02 0.741 0.459213
## Saturationmuted:B:Hue 1.980e-02 3.904e-02 0.507 0.612201
## Saturationneutral:B:Hue -4.266e-02 5.169e-02 -0.825 0.409520
## Saturationpure:B:Hue -7.908e-03 3.794e-02 -0.208 0.834956
## Saturationshaded:B:Hue -5.420e-02 4.629e-02 -1.171 0.242119
## Saturationsubdued:B:Hue 2.367e-02 4.151e-02 0.570 0.568665
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.03175 on 640 degrees of freedom
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.999
## F-statistic: 4260 on 194 and 640 DF, p-value: < 2.2e-16
re_load_mod05 %>% summary()
##
## Call:
## lm(formula = y ~ (Lightness + Saturation) * (R + G + B + Hue),
## data = df_standard)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.218807 -0.029587 -0.002964 0.023051 0.211793
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0156508 0.0333284 0.470 0.638778
## Lightnessdeep -0.0902948 0.0344015 -2.625 0.008844 **
## Lightnesslight -0.3303232 0.0419341 -7.877 1.14e-14 ***
## Lightnessmidtone -0.0813395 0.0341341 -2.383 0.017417 *
## Lightnesspale -0.6291915 0.0528630 -11.902 < 2e-16 ***
## Lightnesssaturated -0.0625005 0.0334789 -1.867 0.062300 .
## Lightnesssoft -0.1545354 0.0356907 -4.330 1.69e-05 ***
## Saturationgray -0.0440442 0.0115286 -3.820 0.000144 ***
## Saturationmuted -0.0291692 0.0063463 -4.596 5.03e-06 ***
## Saturationneutral -0.0618573 0.0067847 -9.117 < 2e-16 ***
## Saturationpure 0.0518827 0.0071504 7.256 9.75e-13 ***
## Saturationshaded -0.0531590 0.0064988 -8.180 1.17e-15 ***
## Saturationsubdued -0.0478015 0.0064215 -7.444 2.62e-13 ***
## R 0.1892266 0.0088042 21.493 < 2e-16 ***
## G 0.6312372 0.0116501 54.183 < 2e-16 ***
## B 0.1602681 0.0173716 9.226 < 2e-16 ***
## Hue -0.0573637 0.0094517 -6.069 2.02e-09 ***
## Lightnessdeep:R 0.0018104 0.0094027 0.193 0.847370
## Lightnesslight:R 0.1036261 0.0244211 4.243 2.47e-05 ***
## Lightnessmidtone:R 0.0151836 0.0133692 1.136 0.256430
## Lightnesspale:R 0.1812149 0.0433573 4.180 3.26e-05 ***
## Lightnesssaturated:R -0.0007776 0.0106664 -0.073 0.941904
## Lightnesssoft:R 0.0492488 0.0186494 2.641 0.008439 **
## Lightnessdeep:G -0.0580356 0.0116771 -4.970 8.25e-07 ***
## Lightnesslight:G 0.3380262 0.0295808 11.427 < 2e-16 ***
## Lightnessmidtone:G 0.0665761 0.0164872 4.038 5.93e-05 ***
## Lightnesspale:G 0.5858475 0.0457749 12.798 < 2e-16 ***
## Lightnesssaturated:G -0.0118009 0.0126143 -0.936 0.349815
## Lightnesssoft:G 0.2086321 0.0225868 9.237 < 2e-16 ***
## Lightnessdeep:B -0.0494322 0.0201075 -2.458 0.014175 *
## Lightnesslight:B -0.0043713 0.0248889 -0.176 0.860630
## Lightnessmidtone:B -0.0644335 0.0201605 -3.196 0.001450 **
## Lightnesspale:B 0.0233714 0.0316768 0.738 0.460855
## Lightnesssaturated:B -0.0469872 0.0202082 -2.325 0.020323 *
## Lightnesssoft:B -0.0545076 0.0210505 -2.589 0.009797 **
## Lightnessdeep:Hue 0.0392576 0.0101835 3.855 0.000125 ***
## Lightnesslight:Hue 0.0525404 0.0098568 5.330 1.29e-07 ***
## Lightnessmidtone:Hue 0.0473387 0.0102896 4.601 4.92e-06 ***
## Lightnesspale:Hue 0.0544998 0.0103466 5.267 1.80e-07 ***
## Lightnesssaturated:Hue 0.0375392 0.0099384 3.777 0.000171 ***
## Lightnesssoft:Hue 0.0534146 0.0103915 5.140 3.48e-07 ***
## Saturationgray:R -0.1580780 0.0675196 -2.341 0.019475 *
## Saturationmuted:R -0.0187794 0.0114883 -1.635 0.102531
## Saturationneutral:R 0.0608227 0.0285038 2.134 0.033171 *
## Saturationpure:R -0.0011462 0.0086715 -0.132 0.894881
## Saturationshaded:R 0.0127518 0.0200588 0.636 0.525149
## Saturationsubdued:R 0.0060838 0.0152335 0.399 0.689732
## Saturationgray:G 0.2688204 0.0762993 3.523 0.000451 ***
## Saturationmuted:G 0.0382669 0.0135997 2.814 0.005021 **
## Saturationneutral:G -0.0058799 0.0331360 -0.177 0.859204
## Saturationpure:G 0.0340289 0.0110360 3.083 0.002119 **
## Saturationshaded:G 0.0410349 0.0237407 1.728 0.084307 .
## Saturationsubdued:G 0.0330834 0.0176914 1.870 0.061859 .
## Saturationgray:B -0.0628830 0.0409159 -1.537 0.124733
## Saturationmuted:B 0.0008769 0.0100322 0.087 0.930373
## Saturationneutral:B 0.0171906 0.0194636 0.883 0.377396
## Saturationpure:B -0.0400486 0.0086671 -4.621 4.48e-06 ***
## Saturationshaded:B 0.0014420 0.0148811 0.097 0.922827
## Saturationsubdued:B 0.0057016 0.0118594 0.481 0.630819
## Saturationgray:Hue 0.0016934 0.0121588 0.139 0.889268
## Saturationmuted:Hue -0.0075430 0.0087080 -0.866 0.386646
## Saturationneutral:Hue 0.0313044 0.0097180 3.221 0.001330 **
## Saturationpure:Hue 0.0137725 0.0085005 1.620 0.105598
## Saturationshaded:Hue 0.0020142 0.0091012 0.221 0.824907
## Saturationsubdued:Hue 0.0027768 0.0094296 0.294 0.768477
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.04872 on 770 degrees of freedom
## Multiple R-squared: 0.9978, Adjusted R-squared: 0.9976
## F-statistic: 5477 on 64 and 770 DF, p-value: < 2.2e-16
You have explored the relationships; next you must consider the UNCERTAINTY on the residual error through Bayesian modeling techniques! #1.Fit 2 Bayesian linear models – one must be the best model from iiA) : model 9 the second must be another model you fit in iiA): model 8 • State why you chose the second model. • You may use the Laplace Approximation approach we used in lecture and the homework assignments. • Alternatively, you may use rstanarm’s stan_lm() or stan_glm() function to fit full Bayesian linear models with syntax like R’s lm(). • Resources to help with rstanarm if you’re interested: • How to Use the rstanarm Package (r-project.org) • Estimating Regularized Linear Models with rstanarm (r-project.org) • Extra examples also provided on Canvas.
• After fitting the 2 models, you must identify the best model. • Which performance metric did you use to make your selection? • Visualize the regression coefficient posterior summary statistics for your best model. • For your best model: Study the posterior UNCERTAINTY on the likelihood noise (residual error), 𝜎. • How does the lm() maximum likelihood estimate (MLE) on 𝜎 relate to the posterior UNCERTAINTY on 𝜎? • Do you feel the posterior is precise or are we quite uncertain about 𝜎?
Xmat_08 <- model.matrix(re_load_mod08, dfii)
Xmat_09 <- model.matrix(re_load_mod09, dfii)
Create information with prior The info_09 list corresponds to the information for model 9, while info_08 corresponds to the information for model 08. Specify the shared prior mean, mu_beta, to be 0, the shared prior standard deviation, tau_beta, as 2. The prior rate parameter on the noise, sigma_rate, is assigned to 1.
info_08 <- list(
yobs = df_standard$y,
design_matrix = Xmat_08,
mu_beta = 0,
tau_beta = 2,
sigma_rate = 1
)
info_09 <- list(
yobs = df_standard$y,
design_matrix = Xmat_09,
mu_beta = 0,
tau_beta = 2,
sigma_rate = 1
)
Define the log-posterior function lm_logpost(). Use the log-transformation on σ, and so we will actually define the log-posterior in terms of the regression coefficients, β, and the unbounded noise parameter, φ=log[σ].
lm_logpost <- function(unknowns, my_info)
{
length_beta <- length(unknowns)-1
beta_v <- unknowns[1:length_beta] %>% as.vector()
lik_varphi <- unknowns[length(unknowns)]
lik_sigma <- exp(lik_varphi)
X <- my_info$design_matrix
# calculate the linear predictor
mu <- X %*% beta_v
# evaluate the log-likelihood
log_lik <- sum(dnorm(x = my_info$yobs, mean = mu, sd = lik_sigma, log = TRUE))
# evaluate the log-prior
log_prior_beta <- sum(dnorm(x = beta_v, mean = my_info$mu_beta, sd = my_info$tau_beta, log = TRUE))
log_prior_sigma <- dexp(x = lik_sigma, rate = my_info$sigma_rate, log = TRUE)
# add the mean trend prior and noise prior together
log_prior <- log_prior_beta + log_prior_sigma
# account for the transformation
log_derive_adjust <- lik_varphi
# sum together
return(log_lik + log_prior + log_derive_adjust)
}
The my_laplace() function is below
my_laplace <- function(start_guess, logpost_func, ...)
{
fit <- optim(start_guess,
logpost_func,
gr = NULL,
...,
method = "BFGS",
hessian = TRUE,
control = list(fnscale = -1, maxit = 1001))
mode <- fit$par
post_var_matrix <- -solve(fit$hessian)
p <- length(mode) # number of unknown parameters
int <- p/2 * log(2*pi) + 0.5*log(det(post_var_matrix)) + logpost_func(mode, ...)
# package all of the results into a list
list(mode = mode,
var_matrix = post_var_matrix,
log_evidence = int,
converge = ifelse(fit$convergence == 0,
"YES",
"NO"),
iter_counts = as.numeric(fit$counts[1]))
}
Fit the Bayesian linear model 08.
init_guess <- rep(0, ncol(Xmat_08)+1)
laplace_quad_08 <- my_laplace(init_guess, lm_logpost, info_08)
laplace_quad_08$converge
## [1] "YES"
laplace_quad_08$log_evidence
## [1] -Inf
Display the posterior mode and posterior standard deviations
cat("posterior mode: ", laplace_quad_08$mode, "\n\n")
## posterior mode: -0.5513629 0.4137779 0.5438713 0.4166352 0.9896795 0.4397414 0.4431632 -0.01954001 -0.01091312 -0.002595044 -0.05529535 -0.0005549269 -0.01760662 0.166577 0.4209329 -0.3788962 0.1283146 0.04685251 0.02766046 -0.1179428 0.001884472 -0.08980574 -0.02363128 0.0282098 -0.1501897 0.03664078 0.07847458 -0.01062842 0.1386229 0.04779992 -0.4103492 0.06526968 0.05186577 0.2204176 -0.2522191 0.2207841 -0.8000247 0.223685 0.06949775 0.481451 0.4586167 0.5103998 0.457466 0.5398372 0.4990376 -0.2002474 -0.1293847 -0.1541937 -0.1591516 -0.1400747 -0.1550027 -0.009295456 -0.07118169 0.005019286 0.2996579 0.001819671 0.01064443 0.06702035 0.3412107 0.09227306 0.2703786 0.06181998 0.2545549 0.1355801 0.2047849 0.1618887 0.2009575 0.161222 0.2058263 -0.003823479 -0.004469975 -0.006330811 -0.004175158 -0.01386251 -0.01743817 -0.01915716 0.002064728 0.06427014 0.03872306 -0.008951481 0.02518748 -0.09967199 0.0004209751 -0.0617041 0.04179359 0.03413876 -0.0001129228 0.133334 0.02621953 0.011243 -0.0380661 -0.02441206 0.0004063613 -0.0001531142 0.0006291646 0.04206827 0.03795517 0.02495524 0.0187025 0.2865317 0.04393491 -0.08814013 0.008547119 -0.09753258 0.04014154 -0.02129725 0.05561047 -0.3963846 0.01603608 -0.06002794 -0.01903479 -0.228901 0.05114366 -0.1041692 -0.01339706 0.04719936 0.007861356 -0.0006670487 -0.02244184 0.01796383 0.01835175 0.002328396 -0.0100891 0.004484463 0.2524113 0.1213733 0.6786898 0.07215723 0.1005221 -0.01312053 -0.06498183 0.003397747 -0.4410236 0.04432404 0.007837539 -0.003116209 0.001980745 -0.006511085 0.122212 -0.01023658 -0.002219608 0.1076572 0.2152946 0.1481963 0.573516 0.1141759 0.1218802 -0.08552693 -0.001482859 -0.01005069 -0.0441347 -0.02260677 -0.04094213 -0.08129615 -0.1350991 -0.08065662 -0.1572569 -0.05638528 -0.08324537 -0.5585624 -0.1146378 0.375611 0.0002207609 0.1805006 -0.07772967 0.1749691 0.05110805 -0.2373467 -0.01693302 -0.05041805 0.01552658 0.0397311 -0.00873217 0.03991291 -0.02223833 -0.04172893 -0.02496097 0.3599958 -0.07757206 0.4439821 0.01259088 -0.03439237 0.04017654 -0.1032538 -0.01785263 -0.02215065 0.00944382 0.07389894 -0.03230595 0.06372182 0.01917394 -0.04205479 -0.00761598 -0.05475638 0.02338721 -3.582063
cat("posterior standard deviations: ", sqrt(diag(laplace_quad_08$var_matrix)))
## posterior standard deviations: 0.1055191 0.1013263 0.1138046 0.106519 0.1520288 0.1035246 0.1070151 0.0282161 0.02708923 0.02677497 0.03914197 0.02698191 0.02703662 0.04224551 0.04829358 0.1117206 0.05619587 0.007982489 0.02507312 0.03349619 0.01575294 0.02680348 0.02335575 0.02766681 0.03720479 0.02971207 0.03481135 0.03762075 0.1193895 0.04136778 0.3095488 0.03806171 0.05417495 0.04375112 0.144329 0.05144059 0.3861441 0.04491607 0.07003661 0.108209 0.1391186 0.1127831 0.2039894 0.1081036 0.1162216 0.05752893 0.06318651 0.05536051 0.07693626 0.05575184 0.05715865 0.007770032 0.06451125 0.01557529 0.1927278 0.008596145 0.03240892 0.01795947 0.1125866 0.03909418 0.2850895 0.0252202 0.06164596 0.03218241 0.05733104 0.03362629 0.093839 0.03321139 0.03915902 0.0104514 0.01466904 0.01372663 0.01391657 0.01163239 0.01500194 0.1131105 0.02455016 0.0460158 0.02646017 0.0348833 0.02885199 0.1332353 0.03858012 0.07850195 0.03430838 0.05728722 0.04660154 0.08544275 0.0276601 0.05140531 0.02480396 0.03822344 0.03249074 0.03223249 0.0259167 0.02671771 0.03235385 0.02639257 0.02639288 0.3444908 0.01230008 0.1272758 0.009164727 0.04199971 0.02279117 0.5127114 0.03037423 0.1867102 0.02430245 0.07916347 0.03931825 0.1920252 0.0198713 0.0677061 0.01349802 0.03950466 0.02317321 0.01660027 0.01341008 0.0141711 0.01223486 0.01307669 0.01368611 0.01649349 0.1155325 0.04126031 0.3112995 0.02441337 0.06608493 0.02081569 0.08964531 0.03112128 0.2392698 0.02435778 0.04809415 0.01513483 0.03907455 0.02276158 0.06818192 0.01779082 0.03013038 0.02687435 0.106475 0.04148795 0.2733178 0.03138539 0.0619568 0.02656696 0.04882158 0.03646567 0.06611806 0.03025403 0.04038842 0.02554426 0.04239234 0.03074304 0.05844103 0.026586 0.03551517 0.7871181 0.03296569 0.277536 0.027661 0.08997499 0.04876403 0.3685498 0.0269305 0.1137736 0.01872412 0.06574094 0.0337582 0.1113503 0.02812207 0.06868927 0.03324323 0.04115023 0.03313617 0.5320436 0.04604977 0.1975533 0.03018513 0.1107034 0.0551565 0.1247469 0.0254364 0.06583479 0.02384029 0.03693237 0.03121329 0.078693 0.03411972 0.04497596 0.03318185 0.04042994 0.03628108 0.024487
Fit the Bayesian linear model 09
init_guess <- rep(0, ncol(Xmat_09)+1)
laplace_quad_09 <- my_laplace(init_guess, lm_logpost, info_09)
laplace_quad_09$converge
## [1] "YES"
laplace_quad_09$log_evidence
## [1] 1218.426
Display the posterior mode and posterior standard deviations
cat("posterior mode: ", laplace_quad_09$mode, "\n\n")
## posterior mode: -0.1657045 0.03532588 0.005869348 0.04562268 0.05738303 0.05023855 0.01902506 0.005425965 0.008165973 -0.00189388 0.02175495 0.003596849 0.0007556507 0.2396139 0.6454933 0.1285493 0.01294464 0.04906861 0.09574578 0.03968679 -0.01621884 0.01773994 -0.005136824 -0.01238986 0.007694969 0.02452262 -0.03187443 0.09941477 -0.04764033 0.01165765 0.03131431 -0.005566811 -3.124398
cat("posterior standard deviations: ", sqrt(diag(laplace_quad_09$var_matrix)))
## posterior standard deviations: 0.0171563 0.008240401 0.01884723 0.01582103 0.01961775 0.01232755 0.01765138 0.008073312 0.005831563 0.006728248 0.006187497 0.006421094 0.006156087 0.007259569 0.01044355 0.009565383 0.005668725 0.004200285 0.00772694 0.0058205 0.003243595 0.008423175 0.007202855 0.009852411 0.008447183 0.007011812 0.009478973 0.004242994 0.006585712 0.008049399 0.005039858 0.003228666 0.02448401
Calculate the posterior model weight
cat("log_evidence model 09: ", laplace_quad_09$log_evidence, "\n\n")
## log_evidence model 09: 1218.426
cat("log_evidence model 08: ", laplace_quad_08$log_evidence, "\n\n")
## log_evidence model 08: -Inf
Model 09 is the best one in Laplace Approximation approach, Residual standard error: 0.04481
varphi_09 <- laplace_quad_09$mode[length(laplace_quad_09$mode)]
cat("posterior UNCERTAINTY model 09: ", exp(varphi_09), "\n")
## posterior UNCERTAINTY model 09: 0.04396338
posterior standard deviations: 0.02448401
Since 0.04481 and 0.04396338 are close, and the standard deviations is quite large compared with posterior uncertainty, we could say that it is uncertain about the σ
##Part ii: Regression – iiC) Linear models Predictions
viz_grid <- expand.grid(R = 0,
G = seq(-3, 3, length.out=75),
B = 0,
Hue = seq(-2.5, 2.5, length.out=6),
#Lightness = unique(df_standard$Lightness),
Lightness = "dark",
#Saturation = unique(df_standard$Saturation),
Saturation = "gray",
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE) %>%
as.data.frame() %>% tibble::as_tibble()
viz_grid %>% glimpse()
## Rows: 450
## Columns: 6
## $ R <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ G <dbl> -3.000000, -2.918919, -2.837838, -2.756757, -2.675676, -2.5…
## $ B <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ Hue <dbl> -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5,…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "gray", "gray", "gray", "gray", "gray", "gray", "gray", "gr…
tidy_predict <- function(mod, xnew)
{
pred_df <- predict(mod, xnew, interval = "confidence") %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred = fit, ci_lwr = lwr, ci_upr = upr) %>%
bind_cols(predict(mod, xnew, interval = 'prediction') %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred_lwr = lwr, pred_upr = upr))
xnew %>% bind_cols(pred_df)
}
#Use non-Bayesian models for the predictions Make predictions with each of the models
pred_lm_09 <- tidy_predict(re_load_mod09, viz_grid)
pred_lm_09 %>% glimpse()
## Rows: 450
## Columns: 11
## $ R <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ G <dbl> -3.000000, -2.918919, -2.837838, -2.756757, -2.675676, -2.5…
## $ B <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ Hue <dbl> -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5, -2.5,…
## $ Lightness <chr> "dark", "dark", "dark", "dark", "dark", "dark", "dark", "da…
## $ Saturation <chr> "gray", "gray", "gray", "gray", "gray", "gray", "gray", "gr…
## $ pred <dbl> -1.1848697, -1.1834536, -1.1807786, -1.1768445, -1.1716515,…
## $ ci_lwr <dbl> -1.3401496, -1.3317272, -1.3222454, -1.3117049, -1.3001065,…
## $ ci_upr <dbl> -1.0295898, -1.0351801, -1.0393118, -1.0419842, -1.0431966,…
## $ pred_lwr <dbl> -1.3633269, -1.3558489, -1.3473557, -1.3378489, -1.3273300,…
## $ pred_upr <dbl> -1.0064126, -1.0110583, -1.0142014, -1.0158402, -1.0159731,…
visualization for each model’s prediction
pred_lm_09 %>% ggplot(aes(x = G)) + geom_ribbon(aes(ymin = pred_lwr, ymax = pred_upr), fill = 'blue') + geom_ribbon(aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(aes(y = pred)) + facet_wrap(~Hue) +
labs(x = "pred_lm_09$x1")
pred_lm_08 <- tidy_predict(re_load_mod08, viz_grid)
compare the visualizations across models include a coord_cartesian() layer with the ylim argument set to c(-2,4)
pred_lm_08 %>% ggplot(aes(x = G)) + geom_ribbon(aes(ymin = pred_lwr, ymax = pred_upr), fill = 'blue') + geom_ribbon(aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(aes(y = pred)) + facet_wrap(~Hue) + coord_cartesian(ylim = c(-2,4)) +
labs(x = "pred_lm_08$x1")
When R < 2 ,the predictive trends are similar between the 2 selected linear models.When R > 2, the predictive trends look different. The uncertainty of model 8 is big, showing that model 8 is over-fitting.
##Part ii: Regression – iiD) Train/tune with resampling • Linear models: • All categorical and continuous inputs - linear additive features • Add categorical inputs to all main effect and all pairwise interactions of continuous inputs • The 2 models selected from iiA) (if they are not one of the two above) • Regularized regression with Elastic net • Add categorical inputs to all main effect and all pairwise interactions of continuous inputs • The more complex of the 2 models selected from iiA) • Neural network • Random forest • Gradient boosted tree • 2 methods of your choice that we did not explicitly discuss in lecture You must use ALL categorical and continuous inputs with the non-linear methods
#specify the resampling scheme and primary performance metric
my_ctrl <- trainControl( method = "repeatedcv", number = 10, repeats = 3)
my_metric <- 'RMSE'
You must train, assess, tune, and compare more complex methods
train_lm_01 <- train(y ~ .,
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)
train_lm_02 <- train(y ~ Lightness + Saturation + (R + G + B + Hue)^2,
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)
train_lm_09 <- train(y ~ Lightness + Saturation + R*G*B*Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2),
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)
train_lm_09$results
## intercept RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 TRUE 0.054563 0.9979538 0.04092225 0.006605376 0.0004091835 0.003634049
train_lm_09 %>% coefplot()
train_lm_08 <- train(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)),
data = dfii,
method = "lm",
metric = my_metric,
preProcess = c("center", "scale"),
trControl = my_ctrl)
train_lm_08$results
## intercept RMSE Rsquared MAE RMSESD RsquaredSD
## 1 TRUE 0.04699707 0.9984425 0.03373372 0.007872184 0.0004992378
## MAESD
## 1 0.003943451
#Regularized regression with Elastic net Train, assess, and tune the glmnet elastic net model with the defined resampling scheme. Assign the result to the enet_default object and display the result to the screen.
set.seed(1234)
enet_default_01 <- caret::train(y ~ Lightness + Saturation + (R + G + B + Hue)^2,
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl)
Create a custom tuning grid to further tune the elastic net lambda and alpha tuning parameters.
lambda_seq <- exp(seq(log(min(enet_default_01$results$lambda)),
log(max(enet_default_01$results$lambda)),
length.out = 25))
enet_grid_01 <- expand.grid(alpha = seq(0.0, 0.15, by = .01), lambda = lambda_seq)
Train, assess, and tune the elastic net model with the custom tuning grid and assign the result to the enet_tune_01 object.
set.seed(1234)
enet_tune_01 <- caret::train(y ~ Lightness + Saturation + (R + G + B + Hue)^2,
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = enet_grid_01)
enet_tune_01$bestTune
## alpha lambda
## 78 0.03 0.003412554
model 09
set.seed(1234)
enet_default_09 <- caret::train(y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl)
enet_default_09$bestTune
## alpha lambda
## 1 0.1 0.002331673
lambda_seq <- exp(seq(log(min(enet_default_09$results$lambda)),
log(max(enet_default_09$results$lambda)),
length.out = 50))
enet_grid_09 <- expand.grid(alpha = seq(0.03, 0.12, by = .01), lambda = lambda_seq)
set.seed(1234)
enet_tune_09 <- caret::train(y ~ Lightness + Saturation + R * G * B * Hue + I(R^2) + I(G^2) + I(B^2) + I(Hue^2),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = enet_grid_09)
enet_tune_09$bestTune
## alpha lambda
## 53 0.04 0.002813845
model 08
set.seed(1234)
enet_default_08 <- caret::train(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl)
enet_default_08$bestTune
## alpha lambda
## 1 0.1 0.002331673
lambda_seq <- exp(seq(log(min(enet_default_08$results$lambda)),
log(max(enet_default_08$results$lambda)),
length.out = 50))
enet_grid_08 <- expand.grid(alpha = seq(0.03, 0.12, by = .01), lambda = lambda_seq)
set.seed(1234)
enet_tune_08 <- caret::train(y ~ (Lightness + Saturation) * (( R + G + B + Hue)^2 + I(R^2) + I(G^2) + I(B^2) + I(Hue^2)),
data = dfii,
method = 'glmnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = enet_grid_08)
enet_tune_08$bestTune
## alpha lambda
## 53 0.04 0.002813845
enet_tune_08$results
## alpha lambda RMSE Rsquared MAE RMSESD RsquaredSD
## 1 0.03 0.002331673 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 2 0.03 0.002561438 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 3 0.03 0.002813845 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 4 0.03 0.003091125 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 5 0.03 0.003395728 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 6 0.03 0.003730347 0.06469133 0.9970976 0.04831645 0.008261880 0.0007151921
## 7 0.03 0.004097939 0.06475364 0.9970925 0.04837121 0.008239364 0.0007131218
## 8 0.03 0.004501755 0.06488398 0.9970815 0.04848669 0.008202778 0.0007098957
## 9 0.03 0.004945363 0.06505613 0.9970668 0.04864269 0.008154320 0.0007057924
## 10 0.03 0.005432685 0.06526198 0.9970494 0.04883199 0.008104553 0.0007009719
## 11 0.03 0.005968028 0.06550002 0.9970292 0.04904412 0.008054690 0.0006955343
## 12 0.03 0.006556124 0.06576161 0.9970069 0.04926500 0.008004048 0.0006898887
## 13 0.03 0.007202172 0.06604178 0.9969829 0.04949372 0.007961838 0.0006848487
## 14 0.03 0.007911882 0.06633855 0.9969571 0.04973025 0.007937011 0.0006814158
## 15 0.03 0.008691528 0.06668191 0.9969264 0.05000907 0.007961344 0.0006835732
## 16 0.03 0.009548001 0.06732573 0.9968678 0.05053250 0.008032438 0.0006923369
## 17 0.03 0.010488872 0.06800293 0.9968067 0.05108916 0.008046858 0.0006951498
## 18 0.03 0.011522457 0.06870501 0.9967414 0.05167610 0.008114937 0.0007030406
## 19 0.03 0.012657894 0.06955303 0.9966615 0.05238087 0.008243988 0.0007174494
## 20 0.03 0.013905217 0.07037429 0.9965833 0.05305938 0.008326494 0.0007278607
## 21 0.03 0.015275453 0.07107525 0.9965162 0.05361570 0.008397099 0.0007360847
## 22 0.03 0.016780713 0.07173281 0.9964529 0.05415292 0.008487554 0.0007449029
## 23 0.03 0.018434304 0.07237017 0.9963911 0.05467640 0.008575698 0.0007529826
## 24 0.03 0.020250841 0.07298426 0.9963314 0.05517455 0.008648617 0.0007597649
## 25 0.03 0.022246382 0.07359752 0.9962713 0.05565850 0.008756934 0.0007699943
## 26 0.03 0.024438566 0.07423341 0.9962088 0.05614753 0.008862872 0.0007795367
## 27 0.03 0.026846770 0.07492348 0.9961405 0.05665576 0.008970189 0.0007898898
## 28 0.03 0.029492282 0.07564245 0.9960688 0.05717438 0.009081163 0.0008011827
## 29 0.03 0.032398485 0.07637916 0.9959949 0.05767260 0.009197069 0.0008127764
## 30 0.03 0.035591069 0.07715558 0.9959171 0.05816467 0.009312992 0.0008243812
## 31 0.03 0.039098253 0.07798333 0.9958341 0.05866270 0.009449550 0.0008368006
## 32 0.03 0.042951040 0.07887029 0.9957448 0.05917442 0.009603052 0.0008511588
## 33 0.03 0.047183484 0.07983058 0.9956480 0.05970632 0.009730927 0.0008645438
## 34 0.03 0.051832999 0.08087562 0.9955425 0.06025165 0.009888193 0.0008793951
## 35 0.03 0.056940682 0.08199837 0.9954283 0.06082387 0.010056117 0.0008967142
## 36 0.03 0.062551681 0.08317393 0.9953086 0.06140897 0.010255160 0.0009182984
## 37 0.03 0.068715596 0.08454000 0.9951672 0.06211804 0.010496599 0.0009450340
## 38 0.03 0.075486909 0.08609745 0.9950036 0.06294213 0.010730461 0.0009737421
## 39 0.03 0.082925475 0.08776279 0.9948269 0.06382311 0.010999936 0.0010062546
## 40 0.03 0.091097046 0.08954004 0.9946373 0.06478202 0.011242468 0.0010382843
## 41 0.03 0.100073853 0.09146241 0.9944310 0.06583223 0.011525035 0.0010759047
## 42 0.03 0.109935245 0.09357598 0.9942013 0.06705153 0.011831727 0.0011171873
## 43 0.03 0.120768389 0.09591907 0.9939421 0.06844893 0.012133769 0.0011633467
## 44 0.03 0.132669044 0.09837418 0.9936692 0.06995364 0.012453260 0.0012111924
## 45 0.03 0.145742403 0.10100331 0.9933730 0.07158695 0.012791449 0.0012629898
## 46 0.03 0.160104026 0.10382156 0.9930526 0.07339517 0.013121470 0.0013170598
## 47 0.03 0.175880860 0.10682344 0.9927072 0.07539176 0.013452182 0.0013754914
## 48 0.03 0.193212361 0.11000161 0.9923388 0.07760260 0.013800554 0.0014358709
## 49 0.03 0.212251727 0.11341051 0.9919403 0.08002810 0.014161212 0.0015000165
## 50 0.03 0.233167255 0.11702212 0.9915150 0.08260372 0.014551578 0.0015711800
## 51 0.04 0.002331673 0.06434578 0.9971271 0.04827285 0.008482254 0.0007185286
## 52 0.04 0.002561438 0.06434578 0.9971271 0.04827285 0.008482254 0.0007185286
## 53 0.04 0.002813845 0.06434578 0.9971271 0.04827285 0.008482254 0.0007185286
## 54 0.04 0.003091125 0.06441981 0.9971211 0.04832502 0.008423029 0.0007142749
## 55 0.04 0.003395728 0.06455125 0.9971105 0.04842608 0.008321999 0.0007069920
## 56 0.04 0.003730347 0.06468372 0.9970998 0.04852912 0.008230164 0.0006999996
## 57 0.04 0.004097939 0.06484274 0.9970868 0.04864502 0.008147612 0.0006932029
## 58 0.04 0.004501755 0.06506585 0.9970682 0.04880313 0.008076719 0.0006869343
## 59 0.04 0.004945363 0.06532835 0.9970459 0.04899824 0.008014006 0.0006813712
## 60 0.04 0.005432685 0.06562305 0.9970205 0.04922407 0.007957927 0.0006768883
## 61 0.04 0.005968028 0.06593262 0.9969936 0.04946291 0.007919397 0.0006740885
## 62 0.04 0.006556124 0.06627152 0.9969634 0.04972765 0.007914773 0.0006757677
## 63 0.04 0.007202172 0.06666023 0.9969278 0.05003858 0.007967327 0.0006844705
## 64 0.04 0.007911882 0.06714814 0.9968834 0.05042398 0.008012910 0.0006911499
## 65 0.04 0.008691528 0.06783967 0.9968204 0.05099654 0.008038191 0.0006966698
## 66 0.04 0.009548001 0.06861943 0.9967475 0.05167032 0.008106574 0.0007067219
## 67 0.04 0.010488872 0.06935469 0.9966783 0.05230262 0.008189705 0.0007157470
## 68 0.04 0.011522457 0.07012753 0.9966054 0.05297301 0.008245447 0.0007223625
## 69 0.04 0.012657894 0.07075826 0.9965450 0.05350044 0.008322799 0.0007301280
## 70 0.04 0.013905217 0.07133317 0.9964897 0.05396697 0.008402139 0.0007375859
## 71 0.04 0.015275453 0.07190271 0.9964346 0.05440702 0.008502945 0.0007455987
## 72 0.04 0.016780713 0.07246828 0.9963794 0.05485642 0.008592201 0.0007540615
## 73 0.04 0.018434304 0.07305463 0.9963219 0.05532690 0.008670970 0.0007610559
## 74 0.04 0.020250841 0.07366406 0.9962617 0.05581267 0.008761015 0.0007696828
## 75 0.04 0.022246382 0.07432874 0.9961956 0.05632453 0.008873265 0.0007798014
## 76 0.04 0.024438566 0.07501312 0.9961273 0.05683575 0.008978684 0.0007896267
## 77 0.04 0.026846770 0.07573066 0.9960559 0.05734978 0.009062715 0.0007975186
## 78 0.04 0.029492282 0.07647109 0.9959815 0.05784901 0.009176572 0.0008091245
## 79 0.04 0.032398485 0.07722404 0.9959056 0.05832678 0.009298081 0.0008209874
## 80 0.04 0.035591069 0.07803870 0.9958234 0.05882226 0.009423555 0.0008323850
## 81 0.04 0.039098253 0.07895688 0.9957298 0.05936800 0.009564534 0.0008461421
## 82 0.04 0.042951040 0.07998569 0.9956239 0.05996086 0.009715191 0.0008610001
## 83 0.04 0.047183484 0.08101113 0.9955192 0.06049000 0.009872476 0.0008773812
## 84 0.04 0.051832999 0.08208157 0.9954097 0.06101179 0.010103233 0.0009011023
## 85 0.04 0.056940682 0.08320820 0.9952954 0.06152979 0.010300118 0.0009222127
## 86 0.04 0.062551681 0.08442055 0.9951726 0.06210111 0.010491548 0.0009401433
## 87 0.04 0.068715596 0.08583528 0.9950259 0.06282401 0.010753382 0.0009677290
## 88 0.04 0.075486909 0.08739317 0.9948632 0.06364379 0.010965116 0.0009900960
## 89 0.04 0.082925475 0.08903411 0.9946904 0.06451960 0.011200444 0.0010185264
## 90 0.04 0.091097046 0.09084925 0.9944988 0.06551424 0.011485257 0.0010519001
## 91 0.04 0.100073853 0.09278169 0.9942949 0.06659743 0.011715324 0.0010810017
## 92 0.04 0.109935245 0.09476850 0.9940848 0.06776117 0.012020034 0.0011163142
## 93 0.04 0.120768389 0.09697151 0.9938495 0.06911267 0.012311410 0.0011532262
## 94 0.04 0.132669044 0.09932890 0.9935954 0.07060709 0.012618902 0.0011945841
## 95 0.04 0.145742403 0.10186992 0.9933197 0.07226225 0.012968620 0.0012419858
## 96 0.04 0.160104026 0.10464836 0.9930151 0.07409249 0.013341021 0.0012969680
## 97 0.04 0.175880860 0.10768922 0.9926777 0.07614422 0.013732890 0.0013582646
## 98 0.04 0.193212361 0.11094358 0.9923138 0.07841133 0.014106815 0.0014231530
## 99 0.04 0.212251727 0.11444213 0.9919178 0.08093087 0.014501349 0.0014937678
## 100 0.04 0.233167255 0.11823264 0.9914837 0.08367532 0.014917187 0.0015720254
## 101 0.05 0.002331673 0.06599745 0.9969818 0.04963519 0.008091300 0.0007098718
## 102 0.05 0.002561438 0.06602428 0.9969798 0.04965176 0.008064375 0.0007077651
## 103 0.05 0.002813845 0.06606825 0.9969761 0.04968186 0.008045238 0.0007065042
## 104 0.05 0.003091125 0.06612997 0.9969709 0.04972540 0.008023859 0.0007044474
## 105 0.05 0.003395728 0.06619682 0.9969653 0.04977131 0.008003177 0.0007022803
## 106 0.05 0.003730347 0.06626809 0.9969593 0.04982267 0.007982522 0.0006998703
## 107 0.05 0.004097939 0.06634565 0.9969528 0.04987700 0.007965617 0.0006977876
## 108 0.05 0.004501755 0.06642619 0.9969459 0.04993325 0.007955470 0.0006960427
## 109 0.05 0.004945363 0.06651275 0.9969384 0.04999205 0.007947710 0.0006944166
## 110 0.05 0.005432685 0.06662503 0.9969287 0.05006444 0.007926683 0.0006915255
## 111 0.05 0.005968028 0.06675902 0.9969171 0.05015266 0.007895290 0.0006881072
## 112 0.05 0.006556124 0.06697096 0.9968984 0.05030813 0.007906626 0.0006876593
## 113 0.05 0.007202172 0.06753036 0.9968473 0.05077383 0.008086236 0.0007012695
## 114 0.05 0.007911882 0.06823725 0.9967822 0.05135319 0.008098162 0.0007025520
## 115 0.05 0.008691528 0.06896331 0.9967149 0.05202077 0.008124520 0.0007073545
## 116 0.05 0.009548001 0.06964351 0.9966501 0.05263200 0.008200716 0.0007168821
## 117 0.05 0.010488872 0.07026779 0.9965911 0.05318459 0.008223317 0.0007203124
## 118 0.05 0.011522457 0.07081931 0.9965381 0.05364650 0.008335262 0.0007294368
## 119 0.05 0.012657894 0.07135380 0.9964865 0.05406004 0.008422962 0.0007364609
## 120 0.05 0.013905217 0.07185256 0.9964380 0.05444180 0.008504849 0.0007435492
## 121 0.05 0.015275453 0.07239734 0.9963847 0.05485109 0.008611972 0.0007536311
## 122 0.05 0.016780713 0.07299284 0.9963261 0.05532754 0.008695805 0.0007603695
## 123 0.05 0.018434304 0.07364642 0.9962615 0.05585836 0.008799177 0.0007693243
## 124 0.05 0.020250841 0.07430506 0.9961960 0.05636927 0.008902381 0.0007795311
## 125 0.05 0.022246382 0.07498425 0.9961283 0.05687850 0.008980316 0.0007872723
## 126 0.05 0.024438566 0.07566466 0.9960606 0.05735333 0.009069558 0.0007952494
## 127 0.05 0.026846770 0.07630035 0.9959972 0.05775526 0.009164997 0.0008044931
## 128 0.05 0.029492282 0.07700627 0.9959266 0.05819048 0.009284266 0.0008154791
## 129 0.05 0.032398485 0.07781705 0.9958446 0.05866912 0.009431321 0.0008291074
## 130 0.05 0.035591069 0.07871840 0.9957527 0.05919032 0.009591654 0.0008451463
## 131 0.05 0.039098253 0.07972159 0.9956497 0.05975180 0.009769208 0.0008636660
## 132 0.05 0.042951040 0.08074327 0.9955451 0.06027815 0.009951459 0.0008821962
## 133 0.05 0.047183484 0.08176965 0.9954407 0.06076243 0.010141032 0.0009003061
## 134 0.05 0.051832999 0.08285937 0.9953301 0.06126446 0.010304600 0.0009159240
## 135 0.05 0.056940682 0.08407021 0.9952066 0.06184138 0.010457395 0.0009299067
## 136 0.05 0.062551681 0.08538090 0.9950721 0.06248948 0.010671923 0.0009518402
## 137 0.05 0.068715596 0.08679432 0.9949266 0.06322470 0.010907073 0.0009764365
## 138 0.05 0.075486909 0.08830705 0.9947706 0.06402878 0.011136104 0.0009998740
## 139 0.05 0.082925475 0.08994124 0.9946005 0.06490908 0.011380280 0.0010265879
## 140 0.05 0.091097046 0.09171479 0.9944153 0.06588035 0.011631348 0.0010555143
## 141 0.05 0.100073853 0.09362466 0.9942153 0.06694782 0.011905996 0.0010878648
## 142 0.05 0.109935245 0.09566984 0.9940013 0.06816869 0.012221274 0.0011242868
## 143 0.05 0.120768389 0.09792021 0.9937634 0.06957544 0.012584526 0.0011690856
## 144 0.05 0.132669044 0.10042109 0.9934948 0.07118747 0.012939611 0.0012174890
## 145 0.05 0.145742403 0.10312171 0.9932019 0.07295712 0.013299653 0.0012699011
## 146 0.05 0.160104026 0.10601626 0.9928860 0.07490088 0.013652811 0.0013246111
## 147 0.05 0.175880860 0.10909823 0.9925479 0.07705657 0.014035496 0.0013853568
## 148 0.05 0.193212361 0.11241231 0.9921815 0.07941359 0.014455983 0.0014558492
## 149 0.05 0.212251727 0.11600648 0.9917814 0.08198735 0.014905317 0.0015346012
## 150 0.05 0.233167255 0.11991603 0.9913420 0.08479580 0.015368550 0.0016200758
## 151 0.06 0.002331673 0.06602278 0.9969886 0.04941966 0.007563829 0.0006431239
## 152 0.06 0.002561438 0.06611745 0.9969797 0.04950308 0.007570636 0.0006450830
## 153 0.06 0.002813845 0.06623065 0.9969693 0.04960837 0.007571432 0.0006463919
## 154 0.06 0.003091125 0.06632374 0.9969605 0.04969565 0.007590679 0.0006500107
## 155 0.06 0.003395728 0.06644922 0.9969487 0.04981232 0.007607122 0.0006531666
## 156 0.06 0.003730347 0.06656775 0.9969376 0.04992154 0.007631107 0.0006570678
## 157 0.06 0.004097939 0.06668564 0.9969265 0.05002891 0.007657050 0.0006612038
## 158 0.06 0.004501755 0.06679490 0.9969161 0.05013256 0.007697403 0.0006669382
## 159 0.06 0.004945363 0.06695141 0.9969014 0.05028095 0.007750499 0.0006733131
## 160 0.06 0.005432685 0.06714537 0.9968834 0.05044851 0.007896301 0.0006854559
## 161 0.06 0.005968028 0.06730687 0.9968680 0.05060760 0.007990098 0.0006955940
## 162 0.06 0.006556124 0.06764251 0.9968365 0.05089391 0.008119790 0.0007071140
## 163 0.06 0.007202172 0.06845384 0.9967618 0.05160845 0.008181530 0.0007118695
## 164 0.06 0.007911882 0.06902702 0.9967090 0.05212574 0.008129115 0.0007103231
## 165 0.06 0.008691528 0.06967411 0.9966471 0.05269914 0.008257297 0.0007212138
## 166 0.06 0.009548001 0.07023437 0.9965941 0.05319590 0.008295085 0.0007254485
## 167 0.06 0.010488872 0.07078441 0.9965412 0.05364472 0.008377449 0.0007321204
## 168 0.06 0.011522457 0.07129681 0.9964913 0.05404522 0.008482651 0.0007408705
## 169 0.06 0.012657894 0.07182049 0.9964399 0.05444373 0.008584286 0.0007501632
## 170 0.06 0.013905217 0.07241434 0.9963812 0.05490674 0.008678387 0.0007585683
## 171 0.06 0.015275453 0.07302105 0.9963215 0.05539766 0.008752197 0.0007646374
## 172 0.06 0.016780713 0.07362670 0.9962614 0.05587272 0.008842403 0.0007733790
## 173 0.06 0.018434304 0.07425578 0.9961986 0.05634394 0.008940841 0.0007831618
## 174 0.06 0.020250841 0.07484683 0.9961398 0.05676153 0.009000271 0.0007896382
## 175 0.06 0.022246382 0.07543054 0.9960819 0.05715132 0.009080600 0.0007969074
## 176 0.06 0.024438566 0.07602455 0.9960232 0.05752811 0.009197609 0.0008072066
## 177 0.06 0.026846770 0.07673839 0.9959513 0.05797472 0.009331806 0.0008203594
## 178 0.06 0.029492282 0.07758495 0.9958654 0.05849610 0.009466136 0.0008336953
## 179 0.06 0.032398485 0.07852664 0.9957692 0.05905735 0.009621772 0.0008490176
## 180 0.06 0.035591069 0.07954945 0.9956639 0.05964251 0.009783569 0.0008659694
## 181 0.06 0.039098253 0.08051941 0.9955652 0.06013775 0.009917051 0.0008783735
## 182 0.06 0.042951040 0.08149988 0.9954658 0.06058246 0.010073395 0.0008915929
## 183 0.06 0.047183484 0.08256780 0.9953569 0.06106842 0.010207269 0.0009041966
## 184 0.06 0.051832999 0.08366278 0.9952462 0.06158260 0.010378338 0.0009208422
## 185 0.06 0.056940682 0.08481650 0.9951303 0.06214566 0.010567508 0.0009360087
## 186 0.06 0.062551681 0.08608842 0.9950018 0.06278100 0.010800635 0.0009569520
## 187 0.06 0.068715596 0.08749228 0.9948588 0.06348639 0.011020333 0.0009790133
## 188 0.06 0.075486909 0.08899714 0.9947051 0.06428019 0.011246827 0.0010014661
## 189 0.06 0.082925475 0.09063926 0.9945360 0.06517749 0.011516199 0.0010308975
## 190 0.06 0.091097046 0.09246986 0.9943456 0.06623117 0.011826741 0.0010651070
## 191 0.06 0.100073853 0.09449794 0.9941328 0.06742253 0.012155910 0.0011037005
## 192 0.06 0.109935245 0.09670719 0.9938998 0.06875474 0.012494715 0.0011442902
## 193 0.06 0.120768389 0.09911346 0.9936435 0.07029468 0.012868094 0.0011917414
## 194 0.06 0.132669044 0.10174091 0.9933598 0.07201514 0.013238154 0.0012423063
## 195 0.06 0.145742403 0.10452543 0.9930577 0.07384551 0.013603040 0.0012974234
## 196 0.06 0.160104026 0.10747476 0.9927377 0.07588663 0.013983449 0.0013578085
## 197 0.06 0.175880860 0.11065581 0.9923923 0.07815043 0.014418624 0.0014265708
## 198 0.06 0.193212361 0.11411765 0.9920133 0.08056964 0.014871175 0.0015022157
## 199 0.06 0.212251727 0.11786257 0.9915997 0.08318840 0.015334734 0.0015853880
## 200 0.06 0.233167255 0.12189773 0.9911518 0.08605963 0.015797891 0.0016745397
## 201 0.07 0.002331673 0.06612137 0.9969781 0.04953104 0.007817640 0.0006656349
## 202 0.07 0.002561438 0.06616965 0.9969737 0.04957278 0.007803131 0.0006651802
## 203 0.07 0.002813845 0.06621951 0.9969691 0.04961458 0.007789929 0.0006648916
## 204 0.07 0.003091125 0.06627128 0.9969643 0.04965654 0.007777268 0.0006647276
## 205 0.07 0.003395728 0.06632642 0.9969592 0.04970156 0.007765852 0.0006646424
## 206 0.07 0.003730347 0.06641224 0.9969512 0.04978470 0.007773234 0.0006654805
## 207 0.07 0.004097939 0.06651188 0.9969417 0.04988280 0.007798032 0.0006684866
## 208 0.07 0.004501755 0.06666180 0.9969277 0.05000662 0.007873719 0.0006752304
## 209 0.07 0.004945363 0.06689524 0.9969044 0.05023632 0.008035583 0.0006950039
## 210 0.07 0.005432685 0.06724710 0.9968712 0.05057148 0.008160660 0.0007088048
## 211 0.07 0.005968028 0.06778992 0.9968226 0.05104959 0.008127031 0.0007098763
## 212 0.07 0.006556124 0.06843141 0.9967644 0.05163841 0.008143265 0.0007082619
## 213 0.07 0.007202172 0.06905435 0.9967057 0.05220010 0.008162190 0.0007132757
## 214 0.07 0.007911882 0.06967379 0.9966471 0.05274207 0.008240185 0.0007206087
## 215 0.07 0.008691528 0.07021778 0.9965951 0.05320744 0.008317084 0.0007270094
## 216 0.07 0.009548001 0.07076055 0.9965425 0.05364820 0.008412619 0.0007349014
## 217 0.07 0.010488872 0.07125137 0.9964947 0.05402042 0.008489388 0.0007423414
## 218 0.07 0.011522457 0.07181447 0.9964394 0.05444734 0.008598158 0.0007521709
## 219 0.07 0.012657894 0.07238322 0.9963834 0.05489409 0.008656014 0.0007573683
## 220 0.07 0.013905217 0.07295714 0.9963265 0.05535403 0.008722185 0.0007634652
## 221 0.07 0.015275453 0.07354572 0.9962676 0.05580146 0.008814622 0.0007721347
## 222 0.07 0.016780713 0.07412682 0.9962097 0.05621426 0.008885228 0.0007788746
## 223 0.07 0.018434304 0.07469651 0.9961530 0.05660842 0.008959421 0.0007861081
## 224 0.07 0.020250841 0.07525537 0.9960977 0.05698338 0.009057574 0.0007956502
## 225 0.07 0.022246382 0.07585182 0.9960387 0.05736943 0.009169251 0.0008066654
## 226 0.07 0.024438566 0.07656039 0.9959678 0.05782690 0.009304066 0.0008185226
## 227 0.07 0.026846770 0.07740513 0.9958821 0.05836042 0.009443917 0.0008317912
## 228 0.07 0.029492282 0.07840265 0.9957790 0.05897594 0.009618125 0.0008491306
## 229 0.07 0.032398485 0.07942030 0.9956742 0.05956770 0.009720600 0.0008604750
## 230 0.07 0.035591069 0.08031375 0.9955837 0.06000058 0.009830362 0.0008697570
## 231 0.07 0.039098253 0.08124529 0.9954889 0.06041402 0.009942743 0.0008798949
## 232 0.07 0.042951040 0.08221867 0.9953905 0.06087064 0.010105133 0.0008929247
## 233 0.07 0.047183484 0.08322177 0.9952898 0.06134719 0.010267893 0.0009062638
## 234 0.07 0.051832999 0.08427383 0.9951848 0.06183901 0.010482143 0.0009238223
## 235 0.07 0.056940682 0.08543029 0.9950702 0.06238168 0.010676276 0.0009397898
## 236 0.07 0.062551681 0.08672574 0.9949410 0.06302533 0.010869751 0.0009564904
## 237 0.07 0.068715596 0.08810992 0.9948017 0.06373288 0.011132968 0.0009813097
## 238 0.07 0.075486909 0.08964941 0.9946452 0.06457386 0.011410884 0.0010077414
## 239 0.07 0.082925475 0.09140004 0.9944636 0.06558534 0.011727221 0.0010416101
## 240 0.07 0.091097046 0.09336444 0.9942577 0.06673917 0.012040668 0.0010773610
## 241 0.07 0.100073853 0.09552366 0.9940290 0.06802582 0.012396760 0.0011198009
## 242 0.07 0.109935245 0.09786501 0.9937800 0.06946474 0.012766315 0.0011643068
## 243 0.07 0.120768389 0.10036740 0.9935118 0.07107692 0.013173796 0.0012193576
## 244 0.07 0.132669044 0.10308972 0.9932171 0.07286831 0.013538150 0.0012721559
## 245 0.07 0.145742403 0.10592823 0.9929121 0.07479546 0.013919656 0.0013293807
## 246 0.07 0.160104026 0.10898381 0.9925838 0.07692166 0.014386505 0.0013992072
## 247 0.07 0.175880860 0.11233348 0.9922192 0.07927386 0.014837174 0.0014743433
## 248 0.07 0.193212361 0.11597074 0.9918209 0.08180541 0.015302054 0.0015570969
## 249 0.07 0.212251727 0.11988129 0.9913917 0.08452779 0.015752251 0.0016430901
## 250 0.07 0.233167255 0.12408848 0.9909292 0.08753082 0.016171684 0.0017320212
## 251 0.08 0.002331673 0.06540228 0.9970398 0.04905608 0.008159286 0.0007038839
## 252 0.08 0.002561438 0.06552516 0.9970292 0.04915486 0.008120710 0.0007001275
## 253 0.08 0.002813845 0.06568983 0.9970150 0.04928526 0.008075543 0.0006958692
## 254 0.08 0.003091125 0.06585061 0.9970009 0.04940661 0.008043689 0.0006928184
## 255 0.08 0.003395728 0.06602215 0.9969856 0.04953473 0.008026308 0.0006909039
## 256 0.08 0.003730347 0.06622089 0.9969680 0.04969136 0.007995982 0.0006879498
## 257 0.08 0.004097939 0.06646926 0.9969462 0.04990221 0.007964757 0.0006849660
## 258 0.08 0.004501755 0.06675436 0.9969202 0.05016849 0.007959621 0.0006857318
## 259 0.08 0.004945363 0.06717827 0.9968793 0.05053837 0.008076469 0.0007035679
## 260 0.08 0.005432685 0.06775667 0.9968256 0.05102232 0.008135539 0.0007109782
## 261 0.08 0.005968028 0.06847941 0.9967605 0.05171246 0.008066147 0.0007036704
## 262 0.08 0.006556124 0.06901161 0.9967098 0.05219179 0.008171773 0.0007168615
## 263 0.08 0.007202172 0.06958915 0.9966556 0.05268474 0.008249955 0.0007212972
## 264 0.08 0.007911882 0.07011654 0.9966054 0.05314027 0.008312999 0.0007260018
## 265 0.08 0.008691528 0.07064633 0.9965537 0.05357187 0.008420835 0.0007353479
## 266 0.08 0.009548001 0.07114233 0.9965053 0.05395102 0.008501990 0.0007424173
## 267 0.08 0.010488872 0.07169173 0.9964513 0.05437073 0.008592301 0.0007504858
## 268 0.08 0.011522457 0.07222679 0.9963986 0.05477859 0.008670318 0.0007578489
## 269 0.08 0.012657894 0.07278954 0.9963424 0.05521174 0.008735262 0.0007643609
## 270 0.08 0.013905217 0.07337058 0.9962843 0.05564600 0.008806676 0.0007716484
## 271 0.08 0.015275453 0.07396058 0.9962250 0.05608067 0.008896602 0.0007808646
## 272 0.08 0.016780713 0.07453183 0.9961676 0.05648016 0.008980839 0.0007898439
## 273 0.08 0.018434304 0.07510653 0.9961106 0.05687358 0.009055951 0.0007971203
## 274 0.08 0.020250841 0.07571244 0.9960507 0.05727809 0.009155105 0.0008067503
## 275 0.08 0.022246382 0.07638252 0.9959842 0.05770316 0.009279810 0.0008179515
## 276 0.08 0.024438566 0.07722920 0.9958982 0.05824720 0.009426767 0.0008304094
## 277 0.08 0.026846770 0.07820104 0.9957980 0.05886034 0.009553771 0.0008434462
## 278 0.08 0.029492282 0.07917899 0.9956969 0.05942298 0.009652648 0.0008543453
## 279 0.08 0.032398485 0.08001575 0.9956122 0.05982200 0.009745541 0.0008616317
## 280 0.08 0.035591069 0.08087638 0.9955253 0.06021482 0.009870375 0.0008719495
## 281 0.08 0.039098253 0.08175600 0.9954372 0.06063088 0.010021810 0.0008835896
## 282 0.08 0.042951040 0.08269281 0.9953437 0.06105697 0.010187084 0.0008961977
## 283 0.08 0.047183484 0.08371247 0.9952419 0.06151452 0.010367445 0.0009104129
## 284 0.08 0.051832999 0.08480353 0.9951337 0.06200826 0.010545794 0.0009242729
## 285 0.08 0.056940682 0.08597817 0.9950175 0.06258245 0.010748156 0.0009403672
## 286 0.08 0.062551681 0.08727611 0.9948886 0.06326164 0.011009019 0.0009623845
## 287 0.08 0.068715596 0.08872837 0.9947421 0.06404774 0.011296236 0.0009888688
## 288 0.08 0.075486909 0.09037166 0.9945732 0.06496592 0.011591625 0.0010187218
## 289 0.08 0.082925475 0.09223233 0.9943793 0.06603872 0.011919396 0.0010546124
## 290 0.08 0.091097046 0.09431118 0.9941596 0.06726403 0.012280611 0.0010963403
## 291 0.08 0.100073853 0.09656979 0.9939190 0.06862568 0.012643001 0.0011407982
## 292 0.08 0.109935245 0.09900091 0.9936592 0.07014717 0.013031878 0.0011905348
## 293 0.08 0.120768389 0.10160167 0.9933806 0.07182908 0.013440095 0.0012441732
## 294 0.08 0.132669044 0.10437139 0.9930837 0.07370031 0.013855710 0.0013023187
## 295 0.08 0.145742403 0.10734614 0.9927645 0.07577738 0.014296918 0.0013683976
## 296 0.08 0.160104026 0.11058474 0.9924140 0.07802877 0.014762359 0.0014426608
## 297 0.08 0.175880860 0.11408562 0.9920335 0.08043973 0.015223017 0.0015219700
## 298 0.08 0.193212361 0.11781969 0.9916285 0.08301066 0.015652759 0.0016027948
## 299 0.08 0.212251727 0.12165344 0.9912224 0.08572912 0.015956923 0.0016705477
## 300 0.08 0.233167255 0.12545817 0.9908456 0.08850224 0.016223528 0.0017290777
## 301 0.09 0.002331673 0.06450885 0.9971127 0.04837535 0.008196725 0.0007038527
## 302 0.09 0.002561438 0.06475829 0.9970917 0.04858027 0.008160155 0.0007022357
## 303 0.09 0.002813845 0.06501201 0.9970710 0.04878145 0.008074917 0.0006964606
## 304 0.09 0.003091125 0.06527306 0.9970491 0.04898692 0.008006119 0.0006926078
## 305 0.09 0.003395728 0.06560081 0.9970204 0.04924579 0.007984734 0.0006935216
## 306 0.09 0.003730347 0.06600725 0.9969846 0.04957253 0.007968207 0.0006939355
## 307 0.09 0.004097939 0.06647671 0.9969428 0.04995438 0.007955978 0.0006944053
## 308 0.09 0.004501755 0.06702724 0.9968931 0.05043546 0.007996410 0.0006985259
## 309 0.09 0.004945363 0.06771564 0.9968301 0.05104785 0.008080197 0.0007051509
## 310 0.09 0.005432685 0.06840034 0.9967668 0.05166704 0.008151336 0.0007120400
## 311 0.09 0.005968028 0.06893095 0.9967170 0.05213453 0.008229353 0.0007193280
## 312 0.09 0.006556124 0.06946373 0.9966667 0.05259205 0.008276857 0.0007239022
## 313 0.09 0.007202172 0.06997689 0.9966182 0.05302494 0.008343250 0.0007287593
## 314 0.09 0.007911882 0.07047994 0.9965696 0.05344150 0.008427799 0.0007357939
## 315 0.09 0.008691528 0.07098573 0.9965200 0.05383792 0.008507092 0.0007429860
## 316 0.09 0.009548001 0.07149538 0.9964697 0.05422345 0.008598959 0.0007511442
## 317 0.09 0.010488872 0.07204353 0.9964155 0.05462864 0.008681927 0.0007590260
## 318 0.09 0.011522457 0.07261606 0.9963588 0.05505809 0.008748189 0.0007655494
## 319 0.09 0.012657894 0.07319392 0.9963011 0.05549559 0.008830261 0.0007736313
## 320 0.09 0.013905217 0.07378927 0.9962412 0.05593664 0.008910537 0.0007818160
## 321 0.09 0.015275453 0.07439274 0.9961806 0.05636674 0.008983524 0.0007895651
## 322 0.09 0.016780713 0.07498028 0.9961220 0.05677520 0.009059795 0.0007975720
## 323 0.09 0.018434304 0.07558461 0.9960618 0.05718110 0.009147477 0.0008061893
## 324 0.09 0.020250841 0.07624365 0.9959962 0.05760335 0.009246672 0.0008150904
## 325 0.09 0.022246382 0.07700945 0.9959191 0.05809050 0.009352480 0.0008243512
## 326 0.09 0.024438566 0.07794587 0.9958228 0.05868649 0.009434784 0.0008319954
## 327 0.09 0.026846770 0.07887639 0.9957266 0.05922913 0.009518783 0.0008414413
## 328 0.09 0.029492282 0.07967560 0.9956455 0.05963326 0.009632989 0.0008517086
## 329 0.09 0.032398485 0.08046040 0.9955670 0.06000470 0.009758007 0.0008605140
## 330 0.09 0.035591069 0.08129369 0.9954837 0.06039580 0.009915162 0.0008725711
## 331 0.09 0.039098253 0.08219812 0.9953935 0.06080047 0.010082662 0.0008854428
## 332 0.09 0.042951040 0.08318572 0.9952951 0.06123359 0.010241693 0.0008970376
## 333 0.09 0.047183484 0.08422230 0.9951924 0.06169823 0.010422126 0.0009110963
## 334 0.09 0.051832999 0.08532880 0.9950830 0.06220846 0.010633694 0.0009266155
## 335 0.09 0.056940682 0.08653543 0.9949634 0.06283832 0.010884056 0.0009459823
## 336 0.09 0.062551681 0.08789117 0.9948280 0.06357285 0.011163940 0.0009697664
## 337 0.09 0.068715596 0.08944376 0.9946699 0.06443223 0.011463905 0.0009983773
## 338 0.09 0.075486909 0.09119400 0.9944888 0.06540613 0.011781453 0.0010311566
## 339 0.09 0.082925475 0.09316648 0.9942813 0.06654565 0.012144104 0.0010714880
## 340 0.09 0.091097046 0.09533704 0.9940508 0.06783612 0.012513782 0.0011145132
## 341 0.09 0.100073853 0.09768910 0.9937997 0.06926453 0.012916491 0.0011644322
## 342 0.09 0.109935245 0.10020104 0.9935306 0.07086426 0.013331664 0.0012168481
## 343 0.09 0.120768389 0.10287267 0.9932448 0.07263564 0.013764766 0.0012745794
## 344 0.09 0.132669044 0.10576896 0.9929333 0.07464046 0.014232043 0.0013424909
## 345 0.09 0.145742403 0.10886791 0.9925993 0.07680946 0.014696916 0.0014148688
## 346 0.09 0.160104026 0.11213695 0.9922486 0.07905130 0.015144322 0.0014911012
## 347 0.09 0.175880860 0.11551114 0.9918936 0.08133330 0.015477327 0.0015540461
## 348 0.09 0.193212361 0.11892057 0.9915550 0.08371203 0.015744789 0.0016111101
## 349 0.09 0.212251727 0.12237427 0.9912345 0.08616955 0.016060028 0.0016674040
## 350 0.09 0.233167255 0.12605835 0.9909060 0.08885904 0.016384702 0.0017229815
## 351 0.10 0.002331673 0.06503981 0.9970679 0.04865440 0.007756012 0.0006672514
## 352 0.10 0.002561438 0.06524769 0.9970496 0.04883565 0.007747857 0.0006695842
## 353 0.10 0.002813845 0.06547366 0.9970294 0.04903602 0.007759370 0.0006739406
## 354 0.10 0.003091125 0.06574939 0.9970050 0.04928570 0.007771806 0.0006782240
## 355 0.10 0.003395728 0.06609144 0.9969747 0.04959712 0.007822428 0.0006843920
## 356 0.10 0.003730347 0.06650349 0.9969380 0.04996992 0.007906292 0.0006920533
## 357 0.10 0.004097939 0.06699175 0.9968943 0.05041511 0.008001629 0.0007004432
## 358 0.10 0.004501755 0.06761647 0.9968381 0.05098900 0.008089839 0.0007080987
## 359 0.10 0.004945363 0.06820530 0.9967840 0.05152636 0.008186193 0.0007161074
## 360 0.10 0.005432685 0.06873230 0.9967348 0.05199447 0.008254574 0.0007224822
## 361 0.10 0.005968028 0.06928405 0.9966831 0.05247647 0.008296608 0.0007267970
## 362 0.10 0.006556124 0.06977639 0.9966362 0.05289698 0.008381846 0.0007339322
## 363 0.10 0.007202172 0.07026389 0.9965891 0.05330272 0.008462654 0.0007407980
## 364 0.10 0.007911882 0.07077470 0.9965393 0.05370269 0.008538173 0.0007471112
## 365 0.10 0.008691528 0.07129029 0.9964886 0.05408441 0.008623062 0.0007546825
## 366 0.10 0.009548001 0.07182245 0.9964364 0.05447459 0.008690736 0.0007609289
## 367 0.10 0.010488872 0.07237986 0.9963815 0.05488364 0.008751294 0.0007663946
## 368 0.10 0.011522457 0.07295106 0.9963246 0.05530719 0.008846978 0.0007751474
## 369 0.10 0.012657894 0.07354229 0.9962654 0.05574241 0.008925256 0.0007828314
## 370 0.10 0.013905217 0.07416631 0.9962026 0.05619523 0.008994579 0.0007910419
## 371 0.10 0.015275453 0.07477966 0.9961413 0.05662372 0.009069875 0.0007994425
## 372 0.10 0.016780713 0.07540290 0.9960794 0.05703814 0.009134662 0.0008056544
## 373 0.10 0.018434304 0.07605873 0.9960140 0.05746715 0.009212637 0.0008128068
## 374 0.10 0.020250841 0.07678546 0.9959411 0.05793566 0.009287513 0.0008190468
## 375 0.10 0.022246382 0.07762355 0.9958557 0.05846612 0.009371357 0.0008256062
## 376 0.10 0.024438566 0.07848122 0.9957670 0.05898964 0.009456095 0.0008341052
## 377 0.10 0.026846770 0.07926386 0.9956874 0.05940741 0.009566884 0.0008446035
## 378 0.10 0.029492282 0.08001696 0.9956119 0.05976847 0.009703059 0.0008552940
## 379 0.10 0.032398485 0.08081854 0.9955320 0.06014681 0.009832328 0.0008641931
## 380 0.10 0.035591069 0.08171702 0.9954419 0.06056320 0.009986313 0.0008761189
## 381 0.10 0.039098253 0.08267462 0.9953465 0.06098410 0.010135911 0.0008866180
## 382 0.10 0.042951040 0.08367018 0.9952472 0.06141811 0.010307593 0.0008993057
## 383 0.10 0.047183484 0.08471245 0.9951441 0.06188965 0.010521699 0.0009145196
## 384 0.10 0.051832999 0.08585847 0.9950302 0.06244517 0.010767122 0.0009330131
## 385 0.10 0.056940682 0.08713496 0.9949026 0.06313441 0.011035481 0.0009548786
## 386 0.10 0.062551681 0.08857866 0.9947565 0.06392688 0.011332501 0.0009820033
## 387 0.10 0.068715596 0.09022328 0.9945877 0.06485192 0.011650893 0.0010122111
## 388 0.10 0.075486909 0.09207501 0.9943952 0.06588969 0.012005197 0.0010485653
## 389 0.10 0.082925475 0.09415511 0.9941755 0.06709475 0.012371965 0.0010891659
## 390 0.10 0.091097046 0.09643176 0.9939321 0.06845965 0.012783992 0.0011391886
## 391 0.10 0.100073853 0.09884153 0.9936753 0.06994466 0.013208549 0.0011916792
## 392 0.10 0.109935245 0.10137125 0.9934050 0.07158686 0.013637361 0.0012464227
## 393 0.10 0.120768389 0.10408596 0.9931152 0.07343918 0.014103642 0.0013106820
## 394 0.10 0.132669044 0.10696850 0.9928076 0.07545967 0.014533747 0.0013771322
## 395 0.10 0.145742403 0.10998404 0.9924927 0.07754147 0.014921862 0.0014408992
## 396 0.10 0.160104026 0.11305549 0.9921826 0.07957776 0.015290439 0.0015027857
## 397 0.10 0.175880860 0.11619236 0.9918834 0.08171315 0.015580004 0.0015560725
## 398 0.10 0.193212361 0.11941857 0.9915934 0.08397107 0.015899402 0.0016076564
## 399 0.10 0.212251727 0.12289838 0.9912893 0.08644771 0.016223454 0.0016609059
## 400 0.10 0.233167255 0.12669266 0.9909694 0.08922393 0.016547997 0.0017167639
## 401 0.11 0.002331673 0.06516620 0.9970601 0.04881280 0.008306366 0.0007013016
## 402 0.11 0.002561438 0.06532288 0.9970469 0.04895033 0.008255644 0.0006972261
## 403 0.11 0.002813845 0.06556478 0.9970263 0.04915471 0.008180650 0.0006920659
## 404 0.11 0.003091125 0.06590468 0.9969959 0.04944915 0.008148009 0.0006940055
## 405 0.11 0.003395728 0.06628979 0.9969605 0.04978977 0.008138925 0.0007001948
## 406 0.11 0.003730347 0.06672013 0.9969207 0.05017185 0.008110257 0.0007039408
## 407 0.11 0.004097939 0.06733848 0.9968645 0.05071754 0.008098311 0.0007053304
## 408 0.11 0.004501755 0.06798312 0.9968053 0.05129767 0.008124815 0.0007087196
## 409 0.11 0.004945363 0.06852457 0.9967547 0.05179516 0.008194474 0.0007164497
## 410 0.11 0.005432685 0.06905446 0.9967051 0.05226617 0.008270843 0.0007231166
## 411 0.11 0.005968028 0.06955288 0.9966579 0.05268879 0.008348804 0.0007300457
## 412 0.11 0.006556124 0.07002656 0.9966123 0.05309169 0.008419214 0.0007359995
## 413 0.11 0.007202172 0.07052419 0.9965639 0.05350132 0.008495278 0.0007420767
## 414 0.11 0.007911882 0.07105006 0.9965123 0.05389878 0.008575131 0.0007495537
## 415 0.11 0.008691528 0.07157899 0.9964604 0.05428566 0.008643783 0.0007554689
## 416 0.11 0.009548001 0.07212537 0.9964066 0.05467395 0.008700322 0.0007605969
## 417 0.11 0.010488872 0.07268681 0.9963509 0.05508328 0.008772934 0.0007673383
## 418 0.11 0.011522457 0.07327605 0.9962920 0.05551497 0.008857333 0.0007757087
## 419 0.11 0.012657894 0.07391812 0.9962274 0.05598645 0.008939114 0.0007849437
## 420 0.11 0.013905217 0.07453996 0.9961652 0.05643063 0.008999598 0.0007920209
## 421 0.11 0.015275453 0.07515794 0.9961038 0.05684328 0.009063295 0.0007981487
## 422 0.11 0.016780713 0.07580193 0.9960397 0.05725948 0.009127217 0.0008039049
## 423 0.11 0.018434304 0.07651250 0.9959682 0.05771611 0.009214459 0.0008107003
## 424 0.11 0.020250841 0.07727969 0.9958905 0.05821004 0.009284957 0.0008164873
## 425 0.11 0.022246382 0.07809345 0.9958071 0.05874195 0.009348323 0.0008231231
## 426 0.11 0.024438566 0.07882121 0.9957331 0.05915324 0.009453580 0.0008328971
## 427 0.11 0.026846770 0.07953314 0.9956612 0.05950076 0.009597692 0.0008445017
## 428 0.11 0.029492282 0.08031295 0.9955832 0.05986991 0.009753176 0.0008561724
## 429 0.11 0.032398485 0.08119716 0.9954946 0.06028878 0.009889577 0.0008656751
## 430 0.11 0.035591069 0.08213463 0.9954006 0.06072304 0.010039064 0.0008764323
## 431 0.11 0.039098253 0.08310164 0.9953038 0.06115719 0.010213305 0.0008891277
## 432 0.11 0.042951040 0.08410432 0.9952043 0.06159661 0.010410423 0.0009027625
## 433 0.11 0.047183484 0.08520211 0.9950946 0.06211397 0.010647860 0.0009201522
## 434 0.11 0.051832999 0.08641972 0.9949726 0.06272754 0.010902741 0.0009411422
## 435 0.11 0.056940682 0.08779802 0.9948335 0.06346731 0.011174140 0.0009632034
## 436 0.11 0.062551681 0.08934097 0.9946756 0.06434204 0.011499617 0.0009932241
## 437 0.11 0.068715596 0.09106788 0.9944977 0.06531850 0.011848973 0.0010276818
## 438 0.11 0.075486909 0.09302863 0.9942918 0.06643280 0.012216816 0.0010651584
## 439 0.11 0.082925475 0.09515060 0.9940673 0.06763720 0.012621861 0.0011107163
## 440 0.11 0.091097046 0.09742672 0.9938247 0.06900956 0.013068137 0.0011649654
## 441 0.11 0.100073853 0.09980608 0.9935709 0.07053985 0.013488343 0.0012184260
## 442 0.11 0.109935245 0.10229584 0.9933107 0.07220797 0.013901142 0.0012735596
## 443 0.11 0.120768389 0.10498273 0.9930309 0.07404798 0.014376332 0.0013423585
## 444 0.11 0.132669044 0.10782978 0.9927381 0.07600020 0.014751022 0.0014001826
## 445 0.11 0.145742403 0.11067766 0.9924566 0.07790230 0.015110342 0.0014563789
## 446 0.11 0.160104026 0.11360037 0.9921843 0.07986176 0.015413320 0.0015059727
## 447 0.11 0.175880860 0.11661112 0.9919201 0.08191891 0.015726247 0.0015533506
## 448 0.11 0.193212361 0.11988393 0.9916409 0.08421548 0.016058948 0.0016041527
## 449 0.11 0.212251727 0.12348457 0.9913423 0.08675920 0.016394496 0.0016576243
## 450 0.11 0.233167255 0.12742178 0.9910252 0.08963382 0.016717016 0.0017132638
## 451 0.12 0.002331673 0.06479223 0.9970910 0.04862001 0.008446027 0.0007124853
## 452 0.12 0.002561438 0.06509334 0.9970652 0.04885646 0.008372732 0.0007088632
## 453 0.12 0.002813845 0.06547524 0.9970320 0.04916363 0.008280633 0.0007057736
## 454 0.12 0.003091125 0.06593987 0.9969909 0.04954239 0.008211528 0.0007047992
## 455 0.12 0.003395728 0.06642913 0.9969478 0.04992908 0.008140355 0.0007014028
## 456 0.12 0.003730347 0.06698459 0.9968975 0.05039443 0.008088852 0.0007014168
## 457 0.12 0.004097939 0.06768893 0.9968324 0.05103262 0.008151196 0.0007099059
## 458 0.12 0.004501755 0.06834310 0.9967713 0.05162021 0.008248120 0.0007194319
## 459 0.12 0.004945363 0.06886102 0.9967231 0.05207007 0.008284846 0.0007226657
## 460 0.12 0.005432685 0.06933889 0.9966781 0.05247885 0.008337511 0.0007267780
## 461 0.12 0.005968028 0.06980403 0.9966335 0.05287235 0.008417258 0.0007334903
## 462 0.12 0.006556124 0.07028255 0.9965868 0.05326613 0.008501593 0.0007408942
## 463 0.12 0.007202172 0.07077851 0.9965385 0.05365048 0.008578979 0.0007479575
## 464 0.12 0.007911882 0.07129552 0.9964881 0.05402778 0.008642524 0.0007539556
## 465 0.12 0.008691528 0.07182116 0.9964364 0.05440741 0.008705673 0.0007598689
## 466 0.12 0.009548001 0.07236735 0.9963823 0.05480469 0.008779029 0.0007667374
## 467 0.12 0.010488872 0.07296251 0.9963230 0.05523576 0.008863961 0.0007751113
## 468 0.12 0.011522457 0.07361923 0.9962571 0.05571979 0.008936951 0.0007832647
## 469 0.12 0.012657894 0.07425756 0.9961931 0.05618583 0.008989759 0.0007895270
## 470 0.12 0.013905217 0.07486538 0.9961327 0.05660397 0.009047524 0.0007953646
## 471 0.12 0.015275453 0.07550703 0.9960688 0.05702850 0.009117910 0.0008017851
## 472 0.12 0.016780713 0.07620566 0.9959986 0.05749082 0.009183600 0.0008075854
## 473 0.12 0.018434304 0.07693841 0.9959243 0.05798177 0.009244903 0.0008135155
## 474 0.12 0.020250841 0.07771095 0.9958456 0.05850460 0.009316912 0.0008209033
## 475 0.12 0.022246382 0.07841716 0.9957742 0.05893778 0.009399911 0.0008285948
## 476 0.12 0.024438566 0.07907716 0.9957080 0.05927377 0.009513211 0.0008367500
## 477 0.12 0.026846770 0.07982885 0.9956326 0.05963365 0.009652903 0.0008469294
## 478 0.12 0.029492282 0.08066933 0.9955485 0.06002896 0.009782545 0.0008552501
## 479 0.12 0.032398485 0.08158127 0.9954569 0.06045730 0.009940490 0.0008661295
## 480 0.12 0.035591069 0.08251622 0.9953630 0.06088260 0.010118892 0.0008789684
## 481 0.12 0.039098253 0.08350587 0.9952641 0.06132070 0.010305768 0.0008917751
## 482 0.12 0.042951040 0.08458191 0.9951558 0.06182225 0.010536806 0.0009096706
## 483 0.12 0.047183484 0.08576629 0.9950360 0.06239459 0.010789274 0.0009295300
## 484 0.12 0.051832999 0.08707946 0.9949029 0.06306844 0.011051237 0.0009503857
## 485 0.12 0.056940682 0.08851951 0.9947564 0.06385951 0.011344541 0.0009749588
## 486 0.12 0.062551681 0.09013005 0.9945909 0.06477504 0.011684382 0.0010070711
## 487 0.12 0.068715596 0.09195833 0.9944006 0.06580143 0.012064799 0.0010448748
## 488 0.12 0.075486909 0.09390122 0.9941976 0.06686789 0.012470614 0.0010873660
## 489 0.12 0.082925475 0.09598942 0.9939771 0.06808842 0.012881682 0.0011355775
## 490 0.12 0.091097046 0.09819616 0.9937466 0.06947717 0.013280114 0.0011848596
## 491 0.12 0.100073853 0.10053681 0.9935042 0.07098914 0.013736060 0.0012423608
## 492 0.12 0.109935245 0.10303917 0.9932460 0.07266875 0.014212332 0.0013071279
## 493 0.12 0.120768389 0.10575410 0.9929693 0.07452010 0.014568632 0.0013585570
## 494 0.12 0.132669044 0.10842215 0.9927093 0.07631379 0.014922301 0.0014122548
## 495 0.12 0.145742403 0.11116631 0.9924577 0.07814492 0.015215743 0.0014541658
## 496 0.12 0.160104026 0.11396442 0.9922158 0.08003361 0.015547309 0.0015019328
## 497 0.12 0.175880860 0.11703626 0.9919590 0.08212413 0.015878738 0.0015496538
## 498 0.12 0.193212361 0.12043172 0.9916820 0.08448505 0.016222032 0.0016009619
## 499 0.12 0.212251727 0.12415858 0.9913860 0.08712476 0.016547582 0.0016536854
## 500 0.12 0.233167255 0.12823752 0.9910719 0.09009004 0.016877171 0.0017108909
## MAESD
## 1 0.005409631
## 2 0.005409631
## 3 0.005409631
## 4 0.005409631
## 5 0.005409631
## 6 0.005409631
## 7 0.005377973
## 8 0.005328718
## 9 0.005270407
## 10 0.005207975
## 11 0.005150800
## 12 0.005100210
## 13 0.005067736
## 14 0.005055014
## 15 0.005099091
## 16 0.005157846
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## 19 0.005241352
## 20 0.005258061
## 21 0.005274202
## 22 0.005305804
## 23 0.005345681
## 24 0.005372690
## 25 0.005432041
## 26 0.005482529
## 27 0.005515124
## 28 0.005534394
## 29 0.005566259
## 30 0.005596479
## 31 0.005636104
## 32 0.005690036
## 33 0.005733849
## 34 0.005806730
## 35 0.005901141
## 36 0.006013286
## 37 0.006170839
## 38 0.006349378
## 39 0.006551668
## 40 0.006755814
## 41 0.007009178
## 42 0.007276577
## 43 0.007545443
## 44 0.007800346
## 45 0.008111317
## 46 0.008438955
## 47 0.008786700
## 48 0.009070244
## 49 0.009347104
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## 51 0.005728048
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## 53 0.005728048
## 54 0.005691648
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## 57 0.005474796
## 58 0.005421382
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## 67 0.005238807
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## 69 0.005272279
## 70 0.005293367
## 71 0.005345191
## 72 0.005386883
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## 74 0.005476554
## 75 0.005525869
## 76 0.005562582
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## 80 0.005702171
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## 82 0.005776889
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## 87 0.006403102
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## 89 0.006790521
## 90 0.007019591
## 91 0.007230097
## 92 0.007483532
## 93 0.007733424
## 94 0.008002690
## 95 0.008318328
## 96 0.008658152
## 97 0.009001421
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## 100 0.010049162
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## 102 0.005237301
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## 105 0.005193303
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## 128 0.005654142
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## 130 0.005747342
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## 132 0.005902000
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## 144 0.008173252
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## 201 0.004993659
## 202 0.004979564
## 203 0.004967672
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## 283 0.006188045
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## 289 0.007165526
## 290 0.007401561
## 291 0.007668121
## 292 0.007950461
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## 295 0.008950812
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## 298 0.009947044
## 299 0.010154261
## 300 0.010352688
## 301 0.005094300
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## 303 0.005028189
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## 312 0.005303673
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## 315 0.005391774
## 316 0.005441549
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## 371 0.005666022
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## 383 0.006265439
## 384 0.006396397
## 385 0.006553766
## 386 0.006726931
## 387 0.006918113
## 388 0.007152820
## 389 0.007403453
## 390 0.007678086
## 391 0.007971498
## 392 0.008321146
## 393 0.008665756
## 394 0.008991843
## 395 0.009274031
## 396 0.009544625
## 397 0.009777553
## 398 0.010008701
## 399 0.010243928
## 400 0.010467814
## 401 0.005646656
## 402 0.005553625
## 403 0.005437730
## 404 0.005360514
## 405 0.005307996
## 406 0.005235328
## 407 0.005210034
## 408 0.005231219
## 409 0.005275222
## 410 0.005327076
## 411 0.005360524
## 412 0.005382878
## 413 0.005422309
## 414 0.005454768
## 415 0.005471555
## 416 0.005489502
## 417 0.005525277
## 418 0.005570949
## 419 0.005606238
## 420 0.005632598
## 421 0.005670504
## 422 0.005685715
## 423 0.005721930
## 424 0.005739544
## 425 0.005732268
## 426 0.005761066
## 427 0.005832549
## 428 0.005909481
## 429 0.005975883
## 430 0.006040471
## 431 0.006100840
## 432 0.006188958
## 433 0.006314353
## 434 0.006449528
## 435 0.006602115
## 436 0.006796917
## 437 0.007011935
## 438 0.007263008
## 439 0.007545796
## 440 0.007844062
## 441 0.008179116
## 442 0.008506911
## 443 0.008837320
## 444 0.009073942
## 445 0.009325063
## 446 0.009566158
## 447 0.009811639
## 448 0.010058993
## 449 0.010292833
## 450 0.010532931
## 451 0.005339752
## 452 0.005296911
## 453 0.005255910
## 454 0.005249416
## 455 0.005232885
## 456 0.005229190
## 457 0.005294380
## 458 0.005341979
## 459 0.005351427
## 460 0.005358708
## 461 0.005395147
## 462 0.005431369
## 463 0.005454653
## 464 0.005471385
## 465 0.005500509
## 466 0.005535455
## 467 0.005585091
## 468 0.005625554
## 469 0.005653086
## 470 0.005680168
## 471 0.005706056
## 472 0.005730967
## 473 0.005744854
## 474 0.005748746
## 475 0.005762252
## 476 0.005803307
## 477 0.005859475
## 478 0.005916020
## 479 0.005976505
## 480 0.006046429
## 481 0.006124487
## 482 0.006238495
## 483 0.006362767
## 484 0.006510739
## 485 0.006679567
## 486 0.006874877
## 487 0.007120624
## 488 0.007402774
## 489 0.007708539
## 490 0.008008162
## 491 0.008343102
## 492 0.008679472
## 493 0.008883685
## 494 0.009126227
## 495 0.009346557
## 496 0.009599318
## 497 0.009867230
## 498 0.010126813
## 499 0.010352763
## 500 0.010607286
set.seed(1234)
nnet_default <- caret::train( y ~ .,
data = dfii,
method = 'nnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
trace = FALSE)
## Warning in nominalTrainWorkflow(x = x, y = y, wts = weights, info = trainInfo,
## : There were missing values in resampled performance measures.
nnet_default$bestTune
## size decay
## 9 5 0.1
nnet_grid <- expand.grid( size = c(5,9,13,17,21),
decay = exp(seq(-6, 0, length.out = 11)))
set.seed(1234)
nnet_tune <- caret::train( y ~ .,
data = dfii,
method = 'nnet',
metric = my_metric,
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = nnet_grid,
trace = FALSE)
nnet_tune$bestTune
## size decay
## 47 21 0.008229747
plot(nnet_tune, xTrans=log)
registerDoParallel(cores=8)
set.seed(1234)
rf_default <- caret::train( y ~ .,
data = dfii,
method = "rf",
trControl = my_ctrl,
metric = my_metric,
importance = TRUE)
rf_default$bestTune
## mtry
## 2 9
rf_default$results
## mtry RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 2 0.22429331 0.9767999 0.16492021 0.025547536 0.0044085567 0.017438581
## 2 9 0.06968765 0.9967131 0.05124449 0.007394860 0.0005828149 0.004599969
## 3 16 0.08476694 0.9949657 0.06163265 0.007351553 0.0008153420 0.005599433
registerDoParallel(cores=8)
set.seed(1234)
rf_grid <- expand.grid(.mtry = (2:20))
rf_tune <- caret::train( y ~ .,
data = dfii,
method = "rf",
trControl = my_ctrl,
tuneGrid = rf_grid,
metric = my_metric,
importance = TRUE)
rf_tune$bestTune
## mtry
## 7 8
plot(rf_tune, xTrans=log)
registerDoParallel(cores=8)
set.seed(1234)
xgb_default <- caret::train(y ~ .,
data = dfii,
method = "xgbTree",
trControl = my_ctrl,
metric = my_metric,
verbosity = 0,
nthread = 1 )
xgb_default$bestTune
## nrounds max_depth eta gamma colsample_bytree min_child_weight subsample
## 33 150 2 0.3 0 0.8 1 0.75
plot(xgb_default)
xgb_grid <- expand.grid(nrounds = seq(100, 2500, by = 300),
max_depth = c(3, 6, 9, 12),
eta = c(0.125, 0.25, 0.5) * xgb_default$bestTune$eta,
gamma = xgb_default$bestTune$gamma,
colsample_bytree = xgb_default$bestTune$colsample_bytree,
min_child_weight = xgb_default$bestTune$min_child_weight,
subsample = xgb_default$bestTune$subsample)
registerDoParallel(cores=8)
set.seed(1234)
xgb_tune <- caret::train( y ~ .,
data = dfii,
method = "xgbTree",
trControl = my_ctrl,
metric = my_metric,
tuneGrid = xgb_grid,
verbosity = 0,
nthread = 1 )
xgb_tune$bestTune
## nrounds max_depth eta gamma colsample_bytree min_child_weight subsample
## 8 2200 3 0.0375 0 0.8 1 0.75
xgb_tune %>% plot()
#SVM
registerDoParallel(cores=8)
set.seed(1234)
svm <- caret::train( y ~ .,
data = dfii,
method = "svmRadial",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
metric = my_metric)
svm$results
## sigma C RMSE Rsquared MAE RMSESD RsquaredSD
## 1 0.04208154 0.25 0.1367684 0.9876522 0.10242319 0.014024510 0.002181577
## 2 0.04208154 0.50 0.1169683 0.9907386 0.08824700 0.010838741 0.001689743
## 3 0.04208154 1.00 0.1014462 0.9930260 0.07756597 0.009445028 0.001392402
## MAESD
## 1 0.009292271
## 2 0.007834135
## 3 0.007083717
registerDoParallel(cores=8)
set.seed(1234)
sigma_values <- c(0.01, 0.03, 0.1, 1)
C_values <- c(0.25, 0.5, 1, 10, 100, 1000)
# Create the tuning grid
svm_grid <- expand.grid(sigma = sigma_values, C = C_values)
svm_tune <- caret::train( y ~ .,
data = dfii,
method = "svmRadial",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = svm_grid,
metric = my_metric)
svm_tune$bestTune
## sigma C
## 5 0.01 100
svm_tune$results
## sigma C RMSE Rsquared MAE RMSESD RsquaredSD
## 1 0.01 0.25 0.12777795 0.9898228 0.09626653 0.015071645 0.0019356865
## 2 0.01 0.50 0.10896819 0.9920011 0.08193742 0.012590203 0.0015834416
## 3 0.01 1.00 0.09720543 0.9934760 0.07370577 0.010194209 0.0012359300
## 4 0.01 10.00 0.08017618 0.9957290 0.06513445 0.006808714 0.0008036555
## 5 0.01 100.00 0.07875694 0.9960889 0.06454914 0.006396148 0.0007903465
## 6 0.01 1000.00 0.07907856 0.9960585 0.06465774 0.006974880 0.0008243659
## 7 0.03 0.25 0.12936595 0.9889298 0.09713662 0.013829632 0.0020025915
## 8 0.03 0.50 0.11306541 0.9913085 0.08532935 0.010809715 0.0015570479
## 9 0.03 1.00 0.09883779 0.9933521 0.07585835 0.009012259 0.0012182063
## 10 0.03 10.00 0.08390817 0.9953818 0.06761251 0.008727717 0.0011020844
## 11 0.03 100.00 0.08335499 0.9954480 0.06720632 0.008360376 0.0010235638
## 12 0.03 1000.00 0.08361050 0.9954225 0.06729382 0.008013786 0.0009804075
## 13 0.10 0.25 0.17644351 0.9804149 0.12992477 0.018769715 0.0036338246
## 14 0.10 0.50 0.14522795 0.9859694 0.10905217 0.014388853 0.0029293035
## 15 0.10 1.00 0.12562099 0.9891770 0.09620281 0.013705532 0.0027085872
## 16 0.10 10.00 0.11202739 0.9912893 0.08594627 0.012756166 0.0022732438
## 17 0.10 100.00 0.11203188 0.9912965 0.08598945 0.012788991 0.0022808364
## 18 0.10 1000.00 0.11203188 0.9912965 0.08598945 0.012788991 0.0022808364
## 19 1.00 0.25 0.53583855 0.8806530 0.37533793 0.052490547 0.0268413572
## 20 1.00 0.50 0.37880698 0.9281586 0.25308602 0.046126011 0.0192836439
## 21 1.00 1.00 0.29744382 0.9480140 0.19900842 0.039281494 0.0158420011
## 22 1.00 10.00 0.28221300 0.9511905 0.18947958 0.037186028 0.0152540469
## 23 1.00 100.00 0.28221300 0.9511905 0.18947958 0.037186028 0.0152540469
## 24 1.00 1000.00 0.28221300 0.9511905 0.18947958 0.037186028 0.0152540469
## MAESD
## 1 0.010352226
## 2 0.009009789
## 3 0.007709334
## 4 0.005415330
## 5 0.005697106
## 6 0.006251034
## 7 0.009341009
## 8 0.008125951
## 9 0.007241491
## 10 0.006776076
## 11 0.006400757
## 12 0.006124480
## 13 0.011388911
## 14 0.008261096
## 15 0.007986662
## 16 0.007575849
## 17 0.007497325
## 18 0.007497325
## 19 0.032414152
## 20 0.025946183
## 21 0.020642183
## 22 0.020141208
## 23 0.020141208
## 24 0.020141208
#PLS
registerDoParallel(cores=8)
set.seed(1234)
pls <- caret::train( y ~ .,
data = dfii,
method = "pls",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
metric = my_metric)
pls$results
## ncomp RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 0.2594890 0.9528249 0.1985486 0.02058900 0.007201300 0.01589521
## 2 2 0.1711272 0.9795025 0.1313449 0.01623436 0.003620325 0.01158483
## 3 3 0.1363831 0.9869397 0.1032318 0.01461153 0.002738435 0.01019301
registerDoParallel(cores=8)
set.seed(1234)
ncomp_values <- 1:10
pls_grid <- expand.grid(ncomp = ncomp_values)
pls_tune <- caret::train( y ~ .,
data = dfii,
method = "pls",
preProcess = c('center', 'scale'),
trControl = my_ctrl,
tuneGrid = pls_grid,
metric = my_metric)
pls_tune$bestTune
## ncomp
## 9 9
pls_tune$results
## ncomp RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 1 0.25948902 0.9528249 0.19854860 0.020589001 0.007201300 0.015895207
## 2 2 0.17112722 0.9795025 0.13134486 0.016234361 0.003620325 0.011584825
## 3 3 0.13638313 0.9869397 0.10323175 0.014611535 0.002738435 0.010193011
## 4 4 0.10206932 0.9927550 0.07769067 0.009472252 0.001344503 0.006968136
## 5 5 0.09567001 0.9936397 0.07096297 0.009358227 0.001257042 0.006499307
## 6 6 0.09316188 0.9939648 0.06942095 0.009363918 0.001195773 0.006103685
## 7 7 0.09229139 0.9940811 0.06838183 0.009404519 0.001201220 0.005987631
## 8 8 0.09100810 0.9942558 0.06736882 0.009045515 0.001147196 0.005989858
## 9 9 0.08916487 0.9944741 0.06714202 0.009046010 0.001097304 0.005961907
## 10 10 0.08918646 0.9944702 0.06718595 0.009105551 0.001106034 0.006023084
caret_acc_compare <- resamples(list(lm_01 = train_lm_01,
lm_02 = train_lm_02,
lm_09 = train_lm_09,
lm_08 = train_lm_08,
enet_01 = enet_tune_01,
enet_09 = enet_tune_09,
enet_08 = enet_tune_08,
nnet_tune = nnet_tune,
rf_default = rf_default,
rf_tune = rf_tune,
xgb_default = xgb_default,
xgb_tune = xgb_tune,
svm_default = svm,
pls_default = pls,
pls_tune = pls_tune))
dotplot(caret_acc_compare, metric = 'RMSE')
According to this question, the model “lm_08” has the lowest RMSE, the model with the lowest RMSE is generally considered to be the best, as it indicates the closest fit to the observed data, so here model “lm_08” is the